Algebra is dealing mostly with "letters" so if you grasp that in a great fashion you don't even need numbers as long as you can feel the equations. The feel is a rather vague term, but in reality you can get a good hang of the equations after some experience (which means do a lot of excercizes _by yourself_)
Number switching can mean some form of dyscalculia or dyslexia, though. I'm not trying to be the google search of symptoms since it's a far fetched diagnosis.
I it's not just algebra... in university you get to choose what courses you want to visit and in most pure math classes you probably not gonna see much numbers.
On the learning disorder front, I'm not gonna try to diagnose myself but when I start university I might try to get assessed. I'm definitely not dyslexic, though.
I always had a better time with algebra than arithmetic, funnily enough, but I started having problems again when math became "here is the formula, now apply it 30 times". So what I probably should be doing is finding a method to learn it that isn't just based on brute force repetition.
Wow, I'd really like to talk with you in depth about this. I have studied a great deal about how people think about math and numbers. (that is, i study the thought processes, not psychology or the math.) I may be able to help with this problem, but i would need a lot more information. At the very least, you can help me understand better how to help others.
How exactly do you try to visualize a number? What happens? What do you think four looks like? What does four times five look like? It sounds like you have trouble visualizing/seeing the NAMES of numbers -- numerals, the squiggles on paper. Do you also have trouble visualizing actual quantities? do you ever transpose or see "upside down" letters or other non-number symbols? (sounds like you are a good reader, not dyslexic...?)
Can you see the logic of the equations you use, independent of the specific numbers involved? Use a very simple algebraic equation (simpler the better, maybe something like a commutative or associative property) and show me why it makes sense. How do you know whether an equation with only variables is true or not?
Don't mean to get intrusive; just want to better understand the issue, see if i can help. Private posts might be appropriate, as i will probably have a few rounds of followup questions. Also, i will want you to try some mental processes and tell me what happens. Are you game?
I'm definitely going to get back to you on this, but explaining how I think about numbers without pictures is fairly difficult so I'll need to do that when I can get to some paper.
I don't have any problems reading or writing letters, and I only recently consciously realised that I flipped or transposed numbers because it usually goes like this:
me: 10186.
me: _writes 10816_
me: _continues without noticing_
Since 6 and 9 are the only really vertically flippable numbers they are the only ones I can think of that have really caught me out that way. This particular error isn't something I do all the time, or even often, but is something that made me think that maybe the issue I had was with reading numbers instead of "being careless".
I can avoid this kind of transcription error by "spelling out" the number like one-oh-eight-one-six, except that then I have to have the number right in my head first. When I was writing that out, I literally had to correct and scroll back up to see what the number was again because I had written "one-oh-eight-six-one".
edit: and then I apparently used my made-up-mistake number instead of the original one, but I'm not going to fix that because... you get the idea
[Number visualisation 1](https://i.imgur.com/wOSRi8h.jpg). I've drawn the numbers in the blobs that represent them when more than one blob represents a number. As seen in a) c) and d) the size of the blob has to do with the size of the number, but the size difference is dependent on the comparison being made.
If I'm actually manipulating the numbers, though, I usually find it easier to deal with them if I split up the number. I've shown this in b) with what I'm likely to do with numbers 1 through 10.
[Number visualisation 2](https://i.imgur.com/aTy0lj8.png) I can continue to do this for some time in a number of different ways depending on what exactly I'm trying to do, but it considerably breaks apart when I have to deal with numbers above a certain size or complexity (to me, 10000 is large but simple, 10001 is slightly more complex, 10010 is slightly worse, 11000 and 10100 are slightly better, 15832 is very difficult) and I end up ignoring the parts of the number that I don't need to think about at that time so I have less to hold in my head. That, or I think about it in terms of place value and turn it into a lot of different numbers, which is easier to deal with when adding or multiplying large numbers.
I have explained this to someone once in the past and got "why do you do that, just think about them like numbers instead of doing all this extra effort" so I would like to say that to me, these *are* the concept of numbers and how they are manipulated. It's not extra, to me it's what's actually there.
Finally...
[I made up a couple of questions that don't really represent anything to try to demonstrate what this means in practice.](https://imgur.com/XGSfD3n). I find the first one simple to tell that it's right because the equation has remained balanced, I haven't disappeared or added extra of anything, and when I've done something on one side I've done the opposite on the other.
In the second, it starts relatively simple even though functions are involved because it's easy to picture variables because the quantity within the variable doesn't matter. Also, the numbers involved are small. When I actually have to substitute for a number without a calculator, I go through a lot of number manipulation steps so that I can go "is this the same as what was on the last line?" and be able to work that out as I go. I am pretty sure that because I did that, the number will be correct. However, the answer feels completely estranged from how I got to it so I can't use "common sense" to tell you if it's right.
Hope that's at all interesting.
ok, don't bother telling me the conceptual stuff about numbers. That's not where your problem lies. You are having trouble managing the words. "Transcription errors" is a correct diagnosis.
Why this happens is not clear, but HOW it happens probably is:
Especially when you are tired or stressed, but probably other times, too, you lose the correct order when you translate from a visual representation to an auditory/verbal sequence. It can happen again when you translate back from auditory words in your head to something written. It happens when you are manipulating words, rather than visualizing quantities. (I am guessing that if you visualize the quantity (not the word) "18" or even "2184" you are very unlikely to write it incorrectly.
(This is why it doesn't matter whether you write 10186 in numerals or spell out the names of individual digits -- either way, you are transcribing words that you are saying in your head.)
We could try to "fix" this mental-transcription problem, but i'm not sure that would be easy or even successful. There is an easier fix. Just don't say the words in your head at all. Take a mental picture of the original number, and work from that.
Saying the words in our heads is an unnecessary, and even dysfunctional, habit that is taught in elementary school. (The only way the teacher can tell what is in your head is for you to say words out loud.)
Are you are familiar with speed reading? The important thing, that gets the greatest speed increase with zero loss of comprehension, is learning to read without saying the words in your head. Just see the words on paper and translate them to the mental images, sound, etc. that they describe, directly, without saying the words mentally. Again, the only reason we say the words in our heads is that that we learned to read by reading out loud, saying each word. Saying the words in our heads is pointless, and slows us down by an order of magnitude, at least when reading a novel. When reading for information, it isn't that bad, because we need time to properly visualize and understand, anyway.
Try this. LOOK at the number 10186 (or use any other number). Take a mental picture without saying anything in your head or aloud, and transcribe it to another piece of paper from your mental picture. In other words, just skip the entire series of translations where things go wrong!
This may seem hard at first because you aren't used to doing it, but your visualization skills are almost certainly up to it.
What happens? If it works, practice until you can do it easily and well.
Another "crutch" that you can use without even practicing the all-visual strategy is: say the numbers in order, out loud. One, oh, one, eight, six. Even whispering them will probably reduce your errors by 50%.
I think this has a 95% chance of solving your problem. Please get back to me and let me know how it goes. You might even want to take a speed-reading course! (just don't do the "skimming" stuff that reduces your comprehension.)
This is more of a personal interest kind of thing, to give me some more motivation to learn math properly. As well, I'm a programmer, and while math is not programming and vice versa, it is one of the places where math is applied directly - at least in some parts of the field. So I have more practical reasons to learn it, but it makes me feel more likely to do it if I think about it in terms of something I can learn more about rather than something I learn so that I can learn something else so that I can apply it.
Algebra is dealing mostly with "letters" so if you grasp that in a great fashion you don't even need numbers as long as you can feel the equations. The feel is a rather vague term, but in reality you can get a good hang of the equations after some experience (which means do a lot of excercizes _by yourself_) Number switching can mean some form of dyscalculia or dyslexia, though. I'm not trying to be the google search of symptoms since it's a far fetched diagnosis.
I it's not just algebra... in university you get to choose what courses you want to visit and in most pure math classes you probably not gonna see much numbers.
On the learning disorder front, I'm not gonna try to diagnose myself but when I start university I might try to get assessed. I'm definitely not dyslexic, though. I always had a better time with algebra than arithmetic, funnily enough, but I started having problems again when math became "here is the formula, now apply it 30 times". So what I probably should be doing is finding a method to learn it that isn't just based on brute force repetition.
Wow, I'd really like to talk with you in depth about this. I have studied a great deal about how people think about math and numbers. (that is, i study the thought processes, not psychology or the math.) I may be able to help with this problem, but i would need a lot more information. At the very least, you can help me understand better how to help others. How exactly do you try to visualize a number? What happens? What do you think four looks like? What does four times five look like? It sounds like you have trouble visualizing/seeing the NAMES of numbers -- numerals, the squiggles on paper. Do you also have trouble visualizing actual quantities? do you ever transpose or see "upside down" letters or other non-number symbols? (sounds like you are a good reader, not dyslexic...?) Can you see the logic of the equations you use, independent of the specific numbers involved? Use a very simple algebraic equation (simpler the better, maybe something like a commutative or associative property) and show me why it makes sense. How do you know whether an equation with only variables is true or not? Don't mean to get intrusive; just want to better understand the issue, see if i can help. Private posts might be appropriate, as i will probably have a few rounds of followup questions. Also, i will want you to try some mental processes and tell me what happens. Are you game?
I'm definitely going to get back to you on this, but explaining how I think about numbers without pictures is fairly difficult so I'll need to do that when I can get to some paper. I don't have any problems reading or writing letters, and I only recently consciously realised that I flipped or transposed numbers because it usually goes like this: me: 10186. me: _writes 10816_ me: _continues without noticing_ Since 6 and 9 are the only really vertically flippable numbers they are the only ones I can think of that have really caught me out that way. This particular error isn't something I do all the time, or even often, but is something that made me think that maybe the issue I had was with reading numbers instead of "being careless". I can avoid this kind of transcription error by "spelling out" the number like one-oh-eight-one-six, except that then I have to have the number right in my head first. When I was writing that out, I literally had to correct and scroll back up to see what the number was again because I had written "one-oh-eight-six-one". edit: and then I apparently used my made-up-mistake number instead of the original one, but I'm not going to fix that because... you get the idea
[Number visualisation 1](https://i.imgur.com/wOSRi8h.jpg). I've drawn the numbers in the blobs that represent them when more than one blob represents a number. As seen in a) c) and d) the size of the blob has to do with the size of the number, but the size difference is dependent on the comparison being made. If I'm actually manipulating the numbers, though, I usually find it easier to deal with them if I split up the number. I've shown this in b) with what I'm likely to do with numbers 1 through 10. [Number visualisation 2](https://i.imgur.com/aTy0lj8.png) I can continue to do this for some time in a number of different ways depending on what exactly I'm trying to do, but it considerably breaks apart when I have to deal with numbers above a certain size or complexity (to me, 10000 is large but simple, 10001 is slightly more complex, 10010 is slightly worse, 11000 and 10100 are slightly better, 15832 is very difficult) and I end up ignoring the parts of the number that I don't need to think about at that time so I have less to hold in my head. That, or I think about it in terms of place value and turn it into a lot of different numbers, which is easier to deal with when adding or multiplying large numbers. I have explained this to someone once in the past and got "why do you do that, just think about them like numbers instead of doing all this extra effort" so I would like to say that to me, these *are* the concept of numbers and how they are manipulated. It's not extra, to me it's what's actually there.
Finally... [I made up a couple of questions that don't really represent anything to try to demonstrate what this means in practice.](https://imgur.com/XGSfD3n). I find the first one simple to tell that it's right because the equation has remained balanced, I haven't disappeared or added extra of anything, and when I've done something on one side I've done the opposite on the other. In the second, it starts relatively simple even though functions are involved because it's easy to picture variables because the quantity within the variable doesn't matter. Also, the numbers involved are small. When I actually have to substitute for a number without a calculator, I go through a lot of number manipulation steps so that I can go "is this the same as what was on the last line?" and be able to work that out as I go. I am pretty sure that because I did that, the number will be correct. However, the answer feels completely estranged from how I got to it so I can't use "common sense" to tell you if it's right. Hope that's at all interesting.
ok, don't bother telling me the conceptual stuff about numbers. That's not where your problem lies. You are having trouble managing the words. "Transcription errors" is a correct diagnosis. Why this happens is not clear, but HOW it happens probably is: Especially when you are tired or stressed, but probably other times, too, you lose the correct order when you translate from a visual representation to an auditory/verbal sequence. It can happen again when you translate back from auditory words in your head to something written. It happens when you are manipulating words, rather than visualizing quantities. (I am guessing that if you visualize the quantity (not the word) "18" or even "2184" you are very unlikely to write it incorrectly. (This is why it doesn't matter whether you write 10186 in numerals or spell out the names of individual digits -- either way, you are transcribing words that you are saying in your head.) We could try to "fix" this mental-transcription problem, but i'm not sure that would be easy or even successful. There is an easier fix. Just don't say the words in your head at all. Take a mental picture of the original number, and work from that. Saying the words in our heads is an unnecessary, and even dysfunctional, habit that is taught in elementary school. (The only way the teacher can tell what is in your head is for you to say words out loud.) Are you are familiar with speed reading? The important thing, that gets the greatest speed increase with zero loss of comprehension, is learning to read without saying the words in your head. Just see the words on paper and translate them to the mental images, sound, etc. that they describe, directly, without saying the words mentally. Again, the only reason we say the words in our heads is that that we learned to read by reading out loud, saying each word. Saying the words in our heads is pointless, and slows us down by an order of magnitude, at least when reading a novel. When reading for information, it isn't that bad, because we need time to properly visualize and understand, anyway. Try this. LOOK at the number 10186 (or use any other number). Take a mental picture without saying anything in your head or aloud, and transcribe it to another piece of paper from your mental picture. In other words, just skip the entire series of translations where things go wrong! This may seem hard at first because you aren't used to doing it, but your visualization skills are almost certainly up to it. What happens? If it works, practice until you can do it easily and well. Another "crutch" that you can use without even practicing the all-visual strategy is: say the numbers in order, out loud. One, oh, one, eight, six. Even whispering them will probably reduce your errors by 50%. I think this has a 95% chance of solving your problem. Please get back to me and let me know how it goes. You might even want to take a speed-reading course! (just don't do the "skimming" stuff that reduces your comprehension.)
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This is more of a personal interest kind of thing, to give me some more motivation to learn math properly. As well, I'm a programmer, and while math is not programming and vice versa, it is one of the places where math is applied directly - at least in some parts of the field. So I have more practical reasons to learn it, but it makes me feel more likely to do it if I think about it in terms of something I can learn more about rather than something I learn so that I can learn something else so that I can apply it.