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I understand that most students understand algebra when it is presented one way, but couldn't teachers give those who don't get it a break and try to explain the process in a different way? If you try to teach me algebra by the example you posted, I am going to be even more confused, and I will bet you've got a few students that are looking at you like you are speaking a foreign language.
Please try to understand that 'isolate the x' doesn't get through to some of us. We need to be shown the process of isolating the x. You have to do the same thing to both sides doesn't make sense to some of us. We have to be able to visualize the process before we understand it.
I guarantee that if you follow my example for your beginning students who don't get it, you will see the smiles turn on, because they will get it. It's just a matter of perspective. You just have to present the materiel in a way students understand. What have you got to loose? A few more students who understand what you are trying to teach? Really, would that be so bad?
Yes but more often than not the student will then confuse parts of each method. The problem is I can't erase what's already in their heads. I find more success with sticking with one method though I have, on rare occasion, retaught a different method when it became apparent that no matter how many times we went over the material this method was not going to work.
When I teach stoichiometry, I teach dimensional analysis because it is so useful and if you get it you can do any problem as long as you know what your starting units and desired units are through a series of conversions and I know that most of my students will get it...eventually. However, every year I have kids who can't do this. Because I know that will be the case, I now open with an exercise that sows the seeds for a second method should I need to come back to it. I find by doing the opening exercise I do, many of the kids who don't get dimensional analysis will actually figure out the other way on their own. I don't teach it because teaching two methods results in confusion of what steps go with which one if the students don't understand. If they understand, they get when they should do which steps.
I have to be very careful to make sure they understand and not just follow an algorithm to get an answer if I teach more than one method. If I don't, they'll do what dd#1 did when she was taught 3 different algorithms for subtraction in everyday mathematics and dump all the steps into a bucket called subtraction and be unable to do anything because they won't know what steps go with which algorithm. If you teach one method and stick to it, you can tell kids to follow the steps who aren't getting it. I find that most of the time, if they do enough problems, the steps start to make sense but we're on dangerous ground if they don't understand and I start introducing a different set of steps.
If I knew in advance which students would get which methods, I would certainly teach different methods to different groups of students. Unfortunately, someone forgot to give me my crystal ball when I became a teacher. I'm guessing at best but I do find that special ed kids get the second version of stoichiometry better than dimensional analysis. I think it's because it breaks what they're doing into intermediate steps.
Last edited by Ivorytickler; 01-03-2014 at 06:44 PM..
Ok, let me see if I can break this down. You teach method A for 3 days. You are still getting the same students raising their hands saying "Ms. Ivory, I don't understand this." THERE'S YOUR SIGN. Give them a little one on one to figure out how they learn and how you can teach them. What I have tried to explain is that some students can not do follow the steps just because some says that is the way it is done. They have to understand why it is done they way it is. For the one's who can look at an equation and get it, you are not going to confuse them if you break the explanation down for those who do not get it. If further explanation confuses them, they did not understand it to began with. Oh, I forgot, it is just easier to fail them and have them hate math and blame them because they say they don't understand it.
Last edited by masonsdaughter; 01-03-2014 at 07:33 PM..
Ok, let me see if I can break this down. You teach method A for 3 days. You are still getting the same students raising their hands saying "Ms. Ivory, I don't understand this." THERE'S YOUR SIGN. Give them a little one on one to figure out how they learn and how you can teach them. Oh, I forgot, it is just easier to fail them and have them hate math and blame them because they say they don't understand it.
Try reading my post. I find greater success by sticking with one method than introducing another because they will mix up the steps to the different methods. Usually, what I will do in a case like this is see if I can get them to break the problem into smaller chunks they can handle easier. My geometry kids really struggle with the fact that problems can be done more than one way. Most of my struggling learners just want an algorithm to follow to get an answer and will get there faster if we stick to one method.
Before becoming a teacher, I thought like you. I really thought that my ability to do problems six different ways including upside down and backwards was an asset. After teaching kids for 6 years now, I've changed my tune. It is very dangerous to introduce multiple methods to kids who don't understand what they are doing because they cannot decide which steps go with which method. They mix up the steps. This is my experience with real students and how they handle being taught multiple methods. Before I try a second method, I'm going to see if explaining the first method in a different way or more practice with the first method will work because I know that if you give kids two ways to do a problem type, they will mix up the steps.
My introduction to this came in my first year in chemistry teaching gas laws. I'd have a problem like X/2 = 6/12 on the board and I'd lose almost every student in the class when I simplified the right side and then multiplied both sides by 2. At first I couldn't figure out what the problem was. In time I realized that to them this is a ratio and they learned to cross multiply and then divide to solve this ratios. So I now cross multiply to get 12X = 6*2 and then divide by 12 to get the x by itself. They're lost if I just multiply both sides by 2. They don't know what I did to solve the problem. Real kids in real classrooms don't handle learning multiple methods of solving problems well. I wish they did. I have all kinds of tricks up my sleeves I could teach them but I know I'd only confuse them.
My experience is that kids learn best learning one method at a time. Teaching a second method requires intensive intervention to avoid confusing the two methods unless they understood the first method. In that case, they can keep the two separate.
Last edited by Ivorytickler; 01-03-2014 at 07:42 PM..
Now I know what is wrong with the educational system. The method I described is the same method used to solve any algebraic equation, it is just a different way of approaching the solution.
Thanks, Ivory, I now know that I wasn't the dumb*** , my teacher was.
I understand that most students understand algebra when it is presented one way, but couldn't teachers give those who don't get it a break and try to explain the process in a different way?
Yes, absolutely. I learned the way in post #179 from our students (alternative ed). I had not learned algebra that way.
Quote:
If you try to teach me algebra by the example you posted, I am going to be even more confused, and I will bet you've got a few students that are looking at you like you are speaking a foreign language.
True. (They actually looked at me that way when I tried to explain it the way I was taught, so I had to find a different way.)
Quote:
Please try to understand that 'isolate the x' doesn't get through to some of us. We need to be shown the process of isolating the x. You have to do the same thing to both sides doesn't make sense to some of us. We have to be able to visualize the process before we understand it.
Yes, and you are probably a visual thinker; geometry, rather than algebra, is probably easier for you. There is a lot of explaining as we go through the steps. Even, "if you learned it this way, this might work better for you...." and I try a whole different example on the board.
I believe, as Ivory said, "The problem is, I can't erase what's already in their heads." This is often what hinders students in learning a different way.
My introduction to this came in my first year in chemistry teaching gas laws. I'd have a problem like X/2 = 6/12 on the board and I'd lose almost every student in the class when I simplified the right side and then multiplied both sides by 2. At first I couldn't figure out what the problem was. In time I realized that to them this is a ratio and they learned to cross multiply and then divide to solve this ratios.
Your students are far beyond ours. Many of our kids would not be able to simplify the right side, especially if we said it that way. Or, "You mean reduce?" some would ask. Others, "You mean make it into a smaller fraction?" So much is semantics and what they have been used to in the past. I no longer tell them to "invert a fraction" when dividing fractions; I say, "flip it" or "turn it upside down."
The problem with teaching only one way to do things is that real life does not always work that way.
One example here with the same problem just given.
40 = 2x + 10
Kids have problems because the variable is on the right side of the equation, but the procedure is still the same one.
40 - 10 = 2x + 10 - 10
30 = 2x
30/2 = 2x/2
15 = x
Computers always place the variable on the left side of the equation and often they tell kids that make the equation with the variable on the right side that they are wrong. They are not wrong, they just changed things around.
Note that when you get into equations with more than one variable, you may have to use both sides of the equation to isolate a single variable. And your answer may include variables, not just numbers.
Your students are far beyond ours. Many of our kids would not be able to simplify the right side, especially if we said it that way. Or, "You mean reduce?" some would ask. Others, "You mean make it into a smaller fraction?" So much is semantics and what they have been used to in the past. I no longer tell them to "invert a fraction" when dividing fractions; I say, "flip it" or "turn it upside down."
Interestingly, while my kids can simplify the right side, they won't until they've completed the cross multiply then divide algorithm. Then and only then will they punch the numbers into their calculators. They have to keep it looking like a ratio.
Most years I've just done cross multiply and divide. This year I'm being belligerent and just saying "This is where you'd cross multiply and divide, but I like to just isolate the x...". I'm trying to see if acknowledging their way of doing it will avoid them becoming confused when I solve the problem in fewer steps.
This is a losing battle though. If I do succeed in getting them to just divide by the 2 to get the x by itself, they'll forego using cross multiply and divide when it's needed as in the problems 2/X = 8/6. They'll divide both sides by 2 and think they isolated the x. The truth is, they make fewer mistakes if I just let them cross multiply and divide all problems that look like ratios.
And yes, my kids are beyond most. They come to me able to do basic algebra. They may need a light review but it doesn't take much. My biggest complaint is the cross multiply and divide thing but that's just a little annoyance because they do get correct answers that way. I'm just lazy and like to take the least number of steps possible. So far, they seem ok with me acknowledging that they would cross multiply and divide in cases where I isolate x. We'll see if this leads to the issue of them not using cross multiply and divide when it's needed. If it does, I think I'll just drink the kool aid and use cross multiply and divide even when it is more steps.
The problem with teaching only one way to do things is that real life does not always work that way.
One example here with the same problem just given.
40 = 2x + 10
Kids have problems because the variable is on the right side of the equation, but the procedure is still the same one.
40 - 10 = 2x + 10 - 10
30 = 2x
30/2 = 2x/2
15 = x
Computers always place the variable on the left side of the equation and often they tell kids that make the equation with the variable on the right side that they are wrong. They are not wrong, they just changed things around.
Note that when you get into equations with more than one variable, you may have to use both sides of the equation to isolate a single variable. And your answer may include variables, not just numbers.
But in real life, you usually master one before you learn another. I find once students master one, they have no issue keeping the steps for each method separate. In fact, once they master one, they often find the others on their own. The problem I have is that if they are confused by one method often adding another method just adds to the confusion. Then we get into trouble.
Got that. Once I understood how equations worked, I could figure out how to do them. It doesn't matter which side the operation is on, once I see how it all fits together, I can see how to do it. I just have to be able to see that I have to do step a because, which leads me to step b because and so on.
Ivory, Now that I think of it, I can sort of understand your position. Your students should have learned how they need to solve problems in lower grades. Not all of them did. Maybe their fault, maybe the fault of a teacher. No matter. A different perspective is not going to confuse those students who know what they are doing. It is not a different way of solving, it is just a different way of seeing the problem.
I hope there is one teacher on this board who can take my posts in to account and have just one student who hates math remember for the rest of his or her life that Mr. or Mrs. Such and Such taught me how to do math and love it. Please. I know there is at least one student in every class who seems hopeless, but who is desperately trying to learn. Give him or her a chance.
I had to do it on my own and it was a struggle, and many times I would give up. I kept trying, though, and finally found out that math is kinda fun. By the way, I was 50+ before I found this out.
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Usually, what I will do in a case like this is see if I can get them to break the problem into smaller chunks they can handle easier. My geometry kids really struggle with the fact that problems can be done more than one way. Most of my struggling learners just want an algorithm to follow to get an answer and will get there faster if we stick to one method. 
