Not terribly long ago, someone asked "Why is it that when you divide by a fraction, you have to turn it upside down and multiply?" I had a lot of trouble trying to figure out how to explain that and a good explanation unraveled from the dialog, both on the list and what she and I e-mailed each other privately. At one point she exclaimed "So, you mean it really is The Same Thing and not some kind of 'trick'?!" And I said "Yes, it really is the same thing." It never occurred to me that anyone would view a shortcut as a 'trick'. I ultimately showed that you can directly divide by a fraction, it is just very convoluted and difficult to keep it all straight in your head, even for me, and that is why this is something where they teach you the Rule in place of teaching you the long process. Usually, schools teach the long process first and only let you use a shorter version if you have thoroughly mastered the long method first.
One example of my own unique thought process is that when you turn the fraction upside down and also change the division sign to a multiplication sign, it is kind of like 'magnetic poles': if you reverse it twice, it is the same thing. Using the reciprocal of the number and also using the reciprocal function (that is probably not an accurate term -- but I mean multiplication and division are 'opposite' in a similar fashion as reciprocals are) is just like reversing the poles on two magnets and thereby having them still do the same thing they were doing -- either attracting or repelling. If you only reverse one magnet, you change what happens. If you reverse both, you get the same result even though it all may be 'turned on its head'.
Another way to think of this would be that if you turn yourself around 180 degrees and face in the opposite direction, and then do it a second time, you will find yourself facing back in the same direction you started with. Using the opposite of the fraction (the reciprocal) and the opposite of the function (multiplication instead of division) gets you turned around twice, so you find yourself still going in the right direction. That is extremely Obvious to me, which is why I had trouble even explaining Why this technique Really is The Same Thing as simply dividing by the original fraction.
I use the concept of 'reciprocals' for a lot of things in math where I have sort of defined my own 'reciprocals' and the 'rules' for how they interact. For example, since we have a base 10 number system, if you think of numbers 1 through 9 as 'reciprocal sets', it makes addition and subtraction much faster. The 'reciprocal' in this case is the number you get when you subtract it from 10. Thus, 1 and 9 are 'reciprocal', 2 and 8, 3 and 7, 4 and 6, and 5 and 5. This helps me enormously because I never really have to add and subtract some things, I just have to know the 'reciprocal'. I do not have to subtract 60 from 100, I can instantly 'see' that the 'reciprocal' of 60 is 40. It is obvious and I don't have to do any calculating. I cannot, at the moment, think of a more complex example but I can tell you that such ideas make me much faster than average at doing math in my head and also mean that I seem to have 'memorized' large amounts of data when I may not really have it memorized, it just takes me a very short time to 'get the answer' because of some Rule I have made up.
I wind up mentally lumping together various 'patterns' that I 'see' that we never got taught were related. All of the rules for how numbers work wind up having some similar rules and some opposite rules and they appear to me to follow a pattern. When you add two even numbers, you get an even number. When you add two odd numbers you get an even number. When you add an even and odd, you get an odd. So, I then tend to think of negative numbers as 'odd' numbers and positive numbers as 'even' numbers and then their rules for multiplying follow the same kind of pattern: 2 negatives gives you a positive product, 2 positives gives you a positive product, and one of each gives you a negative product. So, if you have 2 of the same, you get either even or positive and two different, you get 'odd' numbers (and negative numbers strike me as rather odd creatures indeed, so I feel I have covered that with the word 'odd' without explicitly saying 'negative numbers included'). The word 'Odd' often is used to mean 'different' in our culture, so when two things are different in math, the result is usually 'odd' and when two things are similar the result is the 'norm' (even or positive -- the 'not odd' things in my mind).
Similarly, I group addition and multiplication together as being the same thing and following basically the same rules and I group subtraction and division together and recognize that they largely follow the same kinds of rules as well. For example, both addition and multiplication have commutative properties (it does not matter what order you add or multiply 2 numbers in) and subtraction and division do not (it matters very much which number you are subtracting or dividing by). The fact that I make so many correlations and see a kind of 'grammar' going on here means that I have to learn a lot fewer rules. I learn one rule, I remember what other kinds of things work the same way, and then I only really need to be very clear about the 'exceptions' -- like the ways in which subtraction is different from division or the ways in which negative numbers are not at all 'similar' to odd numbers in how they 'behave' themselves.
Actually, that is a terrible example of an Exception for my way of thinking. In fact, my Default Setting is to group all things that are similar. So, I would be inclined to group traits of negative numbers that are not similar to odd numbers with something completely different rather than continuing to compare them to odd numbers and have to list differences. Listing the differences would have me remembering a bunch more Rules, which would defeat the purpose of thinking about it My Way. So, really, My Way is more like building a web: I knot together the things that are similar and this one 'thread' will be similar to several others and will get 'tied' to each of those other things in the places where they are Close. Exceptions would really be where there is a Gap in my web or maybe a big gnarled knot that keeps Tripping Me Up and I cannot seem to Get Around it without remembering the actual Rule that my math teacher taught me.
Since I do lump together addition and multiplication as being the exact same thing and think of multiplication as 'the shortcut', when I think that even and odd numbers add in the same 'pattern' that negative and positive numbers multiply, I don't even clearly differentiate that these patterns are for different functions. Multiplying is a form of adding so I don't really care about the difference. It is close enough for my logic and for giving me an overview of how all of it hangs together. When I really need to be specific, I can dig around in my brain and come up with the exact rule. The rest of the time, the general principle lets me get by without having a huge catalog of rules uppermost in my mind at all times.
I think that was why Algebra came easily to me and Geometry did not: Short cuts get you in trouble in geometry. They want you to really remember every little rule and write down every little step and I cannot shorten it. Shortening it is actually against the rules. The fact that I mentally make those connections and my mind feels no need whatsoever to remember all the steps -- certain things are 'obvious' to me and I can't imagine why on earth I would have to 'prove' it -- the manner in which I tend to truncate mathematical processes makes it very hard for me to dredge up all the stuff I have 'skipped' and remember every last step so I can put it into a geometric proof. Why on earth would I want to do that?
I often got low marks in geometry when I would shorten a proof from 25 steps to 8 or 10 because the other steps seemed extraneous to me. I guess it would sort of be like reciting the alphabet as "a, b, c, d, l, m, n, o, p, x, y and z" I often just could not see why my teacher insisted that all those other things were necessary. I got to the end, didn't I? And I did it all in the right order, didn't I? And "A comes at the beginning, Z comes at the end and there is some stuff in between. So why are you being so hard on me?" (giggle) Or so I felt at the time.
I also routinely reduce stuff at the beginning rather than the end when working with fractions. This is not the way it is supposed to be done but it is actually much more efficient and I make a lot fewer mistakes, in part because it keeps all the numbers as small as possible. If I have to multiply 1/7 and 7/8, I take the 7 out of both numbers and skip to the end: the answer is obviously 1/8. Why on earth would I want to torture myself by saying it is 7/56 and then trying to find the simplest form? Multiplying it first and then reducing just unnecessarily complicates the whole thing and has you doing 'extra' steps because first you multiply two numbers by 7 and then you divide two numbers by 7. Why bother? Just pull the 7 out of there at the very start and you are much less likely to make a dumb mistake. I mean, I am not going to think it is 1/7 if I reduce first, a mistake I could make if I do it the 'long' way and get confused or distracted for some reason.
I often make dumb mistakes when I make the problem all convoluted by doing it the long way they teach in school. So I simplify first, then calculate. It removes many calculations that are simply 'redundant' in my mind and thus removes additional chances for making a mistake. Often, the shortest route is the one by which you are least likely to get lost, or at least that is how it works for me. I guess it is sort of like that game where one person whispers something in someone's ear and they 'pass it down' and by the time it goes through 20 people it isn't at all what the first person actually said. If I put in all those 'extra' steps, I find that it is much harder to get to the end without making a simple mistake in division, addition or subtraction or some other very basic function.
When doing algebra and above, the vast majority of mistakes are that type of 'dumb' mistake: where you know any kid in elementary school could have performed that particular piece of the problem (multiplied by 7 and then divided by 7, for example) since it is basic arithmetic. But it has you pulling your hair out, trying to figure out where you went wrong, only to notice you forgot to carry the negative sign or you subtracted wrong in the second line of this page-long maze of numbers. To me, the Real Reason those mistakes happen is because these algebra problems take forever. Shortening them gives you better odds of getting through it all with a few hairs still remaining on your head. (This saves on the cost of wigs.)
Anyway, that is a few examples of how I shorten stuff and find 'grammar' and 'syntax' in the language of math. And that is why I sometimes have trouble trying to explain all the steps: I didn't use them to solve the problem. I took a different mental path, and that mental path would, in and of itself, be very hard for me to articulate for a math teacher, if only because it isn't any kind of mathematical rule that my math teacher would admit exists anywhere on planet earth. He certainly never taught me any such thing and what kind of nonsense is this girl spouting? It sounds like fairy tales, not math! It is actually kind of hard for me to really remember the names of the rules, like commutative property, or the difference between natural numbers and whole numbers, etc. But I can work the problems and I actually was an excellent tutor when I was 16 and I have been a godsend for my oldest son, who just never, ever could wrap his brain around math the way they taught it in school but could kind of understand some of mom's explanations.
On his own, he also concluded that reducing fractions at the start is simply more efficient. And I totally agreed, much to his astonishment since he was trying to Justify doing it that way and got no flack from me but had expected to get told he couldn't do that.
Footnote This was written upwards of two decades ago and originally published on an earlier parenting and homeschooling site of mine.
One example of my own unique thought process is that when you turn the fraction upside down and also change the division sign to a multiplication sign, it is kind of like 'magnetic poles': if you reverse it twice, it is the same thing. Using the reciprocal of the number and also using the reciprocal function (that is probably not an accurate term -- but I mean multiplication and division are 'opposite' in a similar fashion as reciprocals are) is just like reversing the poles on two magnets and thereby having them still do the same thing they were doing -- either attracting or repelling. If you only reverse one magnet, you change what happens. If you reverse both, you get the same result even though it all may be 'turned on its head'.
Another way to think of this would be that if you turn yourself around 180 degrees and face in the opposite direction, and then do it a second time, you will find yourself facing back in the same direction you started with. Using the opposite of the fraction (the reciprocal) and the opposite of the function (multiplication instead of division) gets you turned around twice, so you find yourself still going in the right direction. That is extremely Obvious to me, which is why I had trouble even explaining Why this technique Really is The Same Thing as simply dividing by the original fraction.
I use the concept of 'reciprocals' for a lot of things in math where I have sort of defined my own 'reciprocals' and the 'rules' for how they interact. For example, since we have a base 10 number system, if you think of numbers 1 through 9 as 'reciprocal sets', it makes addition and subtraction much faster. The 'reciprocal' in this case is the number you get when you subtract it from 10. Thus, 1 and 9 are 'reciprocal', 2 and 8, 3 and 7, 4 and 6, and 5 and 5. This helps me enormously because I never really have to add and subtract some things, I just have to know the 'reciprocal'. I do not have to subtract 60 from 100, I can instantly 'see' that the 'reciprocal' of 60 is 40. It is obvious and I don't have to do any calculating. I cannot, at the moment, think of a more complex example but I can tell you that such ideas make me much faster than average at doing math in my head and also mean that I seem to have 'memorized' large amounts of data when I may not really have it memorized, it just takes me a very short time to 'get the answer' because of some Rule I have made up.
I wind up mentally lumping together various 'patterns' that I 'see' that we never got taught were related. All of the rules for how numbers work wind up having some similar rules and some opposite rules and they appear to me to follow a pattern. When you add two even numbers, you get an even number. When you add two odd numbers you get an even number. When you add an even and odd, you get an odd. So, I then tend to think of negative numbers as 'odd' numbers and positive numbers as 'even' numbers and then their rules for multiplying follow the same kind of pattern: 2 negatives gives you a positive product, 2 positives gives you a positive product, and one of each gives you a negative product. So, if you have 2 of the same, you get either even or positive and two different, you get 'odd' numbers (and negative numbers strike me as rather odd creatures indeed, so I feel I have covered that with the word 'odd' without explicitly saying 'negative numbers included'). The word 'Odd' often is used to mean 'different' in our culture, so when two things are different in math, the result is usually 'odd' and when two things are similar the result is the 'norm' (even or positive -- the 'not odd' things in my mind).
Similarly, I group addition and multiplication together as being the same thing and following basically the same rules and I group subtraction and division together and recognize that they largely follow the same kinds of rules as well. For example, both addition and multiplication have commutative properties (it does not matter what order you add or multiply 2 numbers in) and subtraction and division do not (it matters very much which number you are subtracting or dividing by). The fact that I make so many correlations and see a kind of 'grammar' going on here means that I have to learn a lot fewer rules. I learn one rule, I remember what other kinds of things work the same way, and then I only really need to be very clear about the 'exceptions' -- like the ways in which subtraction is different from division or the ways in which negative numbers are not at all 'similar' to odd numbers in how they 'behave' themselves.
Actually, that is a terrible example of an Exception for my way of thinking. In fact, my Default Setting is to group all things that are similar. So, I would be inclined to group traits of negative numbers that are not similar to odd numbers with something completely different rather than continuing to compare them to odd numbers and have to list differences. Listing the differences would have me remembering a bunch more Rules, which would defeat the purpose of thinking about it My Way. So, really, My Way is more like building a web: I knot together the things that are similar and this one 'thread' will be similar to several others and will get 'tied' to each of those other things in the places where they are Close. Exceptions would really be where there is a Gap in my web or maybe a big gnarled knot that keeps Tripping Me Up and I cannot seem to Get Around it without remembering the actual Rule that my math teacher taught me.
Since I do lump together addition and multiplication as being the exact same thing and think of multiplication as 'the shortcut', when I think that even and odd numbers add in the same 'pattern' that negative and positive numbers multiply, I don't even clearly differentiate that these patterns are for different functions. Multiplying is a form of adding so I don't really care about the difference. It is close enough for my logic and for giving me an overview of how all of it hangs together. When I really need to be specific, I can dig around in my brain and come up with the exact rule. The rest of the time, the general principle lets me get by without having a huge catalog of rules uppermost in my mind at all times.
I think that was why Algebra came easily to me and Geometry did not: Short cuts get you in trouble in geometry. They want you to really remember every little rule and write down every little step and I cannot shorten it. Shortening it is actually against the rules. The fact that I mentally make those connections and my mind feels no need whatsoever to remember all the steps -- certain things are 'obvious' to me and I can't imagine why on earth I would have to 'prove' it -- the manner in which I tend to truncate mathematical processes makes it very hard for me to dredge up all the stuff I have 'skipped' and remember every last step so I can put it into a geometric proof. Why on earth would I want to do that?
I often got low marks in geometry when I would shorten a proof from 25 steps to 8 or 10 because the other steps seemed extraneous to me. I guess it would sort of be like reciting the alphabet as "a, b, c, d, l, m, n, o, p, x, y and z" I often just could not see why my teacher insisted that all those other things were necessary. I got to the end, didn't I? And I did it all in the right order, didn't I? And "A comes at the beginning, Z comes at the end and there is some stuff in between. So why are you being so hard on me?" (giggle) Or so I felt at the time.
I also routinely reduce stuff at the beginning rather than the end when working with fractions. This is not the way it is supposed to be done but it is actually much more efficient and I make a lot fewer mistakes, in part because it keeps all the numbers as small as possible. If I have to multiply 1/7 and 7/8, I take the 7 out of both numbers and skip to the end: the answer is obviously 1/8. Why on earth would I want to torture myself by saying it is 7/56 and then trying to find the simplest form? Multiplying it first and then reducing just unnecessarily complicates the whole thing and has you doing 'extra' steps because first you multiply two numbers by 7 and then you divide two numbers by 7. Why bother? Just pull the 7 out of there at the very start and you are much less likely to make a dumb mistake. I mean, I am not going to think it is 1/7 if I reduce first, a mistake I could make if I do it the 'long' way and get confused or distracted for some reason.
I often make dumb mistakes when I make the problem all convoluted by doing it the long way they teach in school. So I simplify first, then calculate. It removes many calculations that are simply 'redundant' in my mind and thus removes additional chances for making a mistake. Often, the shortest route is the one by which you are least likely to get lost, or at least that is how it works for me. I guess it is sort of like that game where one person whispers something in someone's ear and they 'pass it down' and by the time it goes through 20 people it isn't at all what the first person actually said. If I put in all those 'extra' steps, I find that it is much harder to get to the end without making a simple mistake in division, addition or subtraction or some other very basic function.
When doing algebra and above, the vast majority of mistakes are that type of 'dumb' mistake: where you know any kid in elementary school could have performed that particular piece of the problem (multiplied by 7 and then divided by 7, for example) since it is basic arithmetic. But it has you pulling your hair out, trying to figure out where you went wrong, only to notice you forgot to carry the negative sign or you subtracted wrong in the second line of this page-long maze of numbers. To me, the Real Reason those mistakes happen is because these algebra problems take forever. Shortening them gives you better odds of getting through it all with a few hairs still remaining on your head. (This saves on the cost of wigs.)
Anyway, that is a few examples of how I shorten stuff and find 'grammar' and 'syntax' in the language of math. And that is why I sometimes have trouble trying to explain all the steps: I didn't use them to solve the problem. I took a different mental path, and that mental path would, in and of itself, be very hard for me to articulate for a math teacher, if only because it isn't any kind of mathematical rule that my math teacher would admit exists anywhere on planet earth. He certainly never taught me any such thing and what kind of nonsense is this girl spouting? It sounds like fairy tales, not math! It is actually kind of hard for me to really remember the names of the rules, like commutative property, or the difference between natural numbers and whole numbers, etc. But I can work the problems and I actually was an excellent tutor when I was 16 and I have been a godsend for my oldest son, who just never, ever could wrap his brain around math the way they taught it in school but could kind of understand some of mom's explanations.
On his own, he also concluded that reducing fractions at the start is simply more efficient. And I totally agreed, much to his astonishment since he was trying to Justify doing it that way and got no flack from me but had expected to get told he couldn't do that.
Footnote This was written upwards of two decades ago and originally published on an earlier parenting and homeschooling site of mine.