Quick post to share the teaching idea. The preservice high school teachers read
“Advice for Solving
Equations,” Steuben and Torbert, Mathematics
Teacher, April 2006
The reflection was to give your advice. While I think the advice they give is solid, what I like is the giving of advice.
John G:
1) remember the meaning of the =. These things are the same thing, though they look different.
2) remember your purpose and for what you’re solving.
3) think about the best representation for solving this equation.
Ellen B:
1. Don’t forget about substitution! I wouldn’t have thought about this in the problem in the reading, but it made the problem so much easier.
2. Could you change the form of the equation to make it easier to solve?
3. Know what you’re solving for; what should the solution be in terms of?
Alyssa B:
1. Look at the problem before you start; what do you already know or what is important to note?
2. Make sure your answers make mathematical sense!
3. If you’re stuck, try rearranging/changing the form of the equation.
Greg O:
1. Never give up on a problem, if your stuck try rearranging it to make more sense to you.
2. Always look for things that you know in a problem so that you can maybe substitute something for it.
3. Make sure you can reverse the process so that you can get the original answer, a way of checking to make sure your answer is correct.
Jordan D:
1. It’s alright if you can’t get it right at first. You learn by spending time on the equation and from mistakes.
2. Experience helps you solve challenging equations, there are many steps that do not seem natural until we use them.
3.Always check your solution, make sure that it makes sense and works.
Emily W:
1. “You will get much more out of a problem if you work on it for 15 to 20 minutes and fail than if you turn to the solution after only 3 minutes.”
2. When you first start a problem, notice everything you can about the problem (Can x = 0? How many solutions will this equation give? Etc.).
3. If the method you are comfortable with does not come up with the correct solution, don’t be afraid to use one you are less comfortable with.
Mike Simon
1.) When solving a problem that looks difficult, begin by stating what you know.
2.) Work with what you know and if you get to a point where you are not sure if there is a next step, look over your work.
3.) After checking over your work, a make a guess and see where it leads you. There is not shame in being wrong.
Matt York
1. The whole purpose of the problem is defeated when you give the solution after just 3 minutes.
2. As you are solving, always keep in mind the goal... never forget the purpose of the problem.
3. If you have no idea where to start, try some numbers which make sense in the context of the problem, it might lead you somewhere.
Joe Freiman
1. It’s better to work for 15 minutes and fail, than to look at the answer after a minute.
2. Using your past experience is the best way to solve difficult equations, so by practicing constantly and gaining more experience is the best way to get better at solving complex equations.
3. Check your work, make sure your answer makes sense.
EDIT: some end-of-the-semester catch-up-additions...
Brandi Stewart
1) It is ok to be wrong and have to try many different ways.
2) Check your work
3) It is not bad to ask for help, especially if you have tried the problem on your own and do not know how to figure it out.
Courtney Johnson
Mine are kind of specific but...
1) Try putting all variables on one side and setting equal to zero to see if using the quadratic formula is a possibilty
2) See if any substitutions can be made (e.g. trig identities)
3) Try removing a common factor to simplify
Amanda Hoezee
1) You will get much more out of a problem if you work on it for15 to 20 minutes and fail than if you turn to the solution after only 3 minutes.
2) One does not need to be talented to solve challenging equations; one only needs experience.
Mitchell Brady
1) If you are stuck on a problem keep it simple and work with what you already know about the problem a solution that may help.
2) If you are stuck go back through your work to see if something sparks your memory to continue.
3) Finally, do not be afraid to continue and be wrong, just learn from your mistakes and try another method, but first figure out why it was wrong so you do not make the same mistake twice.
Ryan Warner
1) Failing isn’t always the worst thing, you can learn a lot sometimes by failing
2) There is almost always more than one way to solve a math problem, when solving equations this is definitely true so try different ways if you can’t figure it out right away.
3) Learn everything you can about the equation first before you try to solve it.
Shannon Penix
1) It is important to look at the problem all the way through before trying to solve it, noticing things you already know.
2) It is alright to take a while to get through a problem and even not succeed. At least there is learning involved.
3) Take the time to go back over your work if you get stuck. There may be earlier steps that could trigger your thinking to continue.
Jeremy Sheaffer
1) Be fair to both sides of the equation.
2) Check your work
3) The harder something is to learn, the more chance there is that you will remember it.
1) remember the meaning of the =. These things are the same thing, though they look different.
2) remember your purpose and for what you’re solving.
3) think about the best representation for solving this equation.
Ellen B:
1. Don’t forget about substitution! I wouldn’t have thought about this in the problem in the reading, but it made the problem so much easier.
2. Could you change the form of the equation to make it easier to solve?
3. Know what you’re solving for; what should the solution be in terms of?
Alyssa B:
1. Look at the problem before you start; what do you already know or what is important to note?
2. Make sure your answers make mathematical sense!
3. If you’re stuck, try rearranging/changing the form of the equation.
Greg O:
1. Never give up on a problem, if your stuck try rearranging it to make more sense to you.
2. Always look for things that you know in a problem so that you can maybe substitute something for it.
3. Make sure you can reverse the process so that you can get the original answer, a way of checking to make sure your answer is correct.
Jordan D:
1. It’s alright if you can’t get it right at first. You learn by spending time on the equation and from mistakes.
2. Experience helps you solve challenging equations, there are many steps that do not seem natural until we use them.
3.Always check your solution, make sure that it makes sense and works.
Emily W:
1. “You will get much more out of a problem if you work on it for 15 to 20 minutes and fail than if you turn to the solution after only 3 minutes.”
2. When you first start a problem, notice everything you can about the problem (Can x = 0? How many solutions will this equation give? Etc.).
3. If the method you are comfortable with does not come up with the correct solution, don’t be afraid to use one you are less comfortable with.
Mike Simon
1.) When solving a problem that looks difficult, begin by stating what you know.
2.) Work with what you know and if you get to a point where you are not sure if there is a next step, look over your work.
3.) After checking over your work, a make a guess and see where it leads you. There is not shame in being wrong.
Matt York
1. The whole purpose of the problem is defeated when you give the solution after just 3 minutes.
2. As you are solving, always keep in mind the goal... never forget the purpose of the problem.
3. If you have no idea where to start, try some numbers which make sense in the context of the problem, it might lead you somewhere.
Joe Freiman
1. It’s better to work for 15 minutes and fail, than to look at the answer after a minute.
2. Using your past experience is the best way to solve difficult equations, so by practicing constantly and gaining more experience is the best way to get better at solving complex equations.
3. Check your work, make sure your answer makes sense.
EDIT: some end-of-the-semester catch-up-additions...
Brandi Stewart
1) It is ok to be wrong and have to try many different ways.
2) Check your work
3) It is not bad to ask for help, especially if you have tried the problem on your own and do not know how to figure it out.
Courtney Johnson
Mine are kind of specific but...
1) Try putting all variables on one side and setting equal to zero to see if using the quadratic formula is a possibilty
2) See if any substitutions can be made (e.g. trig identities)
3) Try removing a common factor to simplify
Amanda Hoezee
1) You will get much more out of a problem if you work on it for15 to 20 minutes and fail than if you turn to the solution after only 3 minutes.
2) One does not need to be talented to solve challenging equations; one only needs experience.
Mitchell Brady
1) If you are stuck on a problem keep it simple and work with what you already know about the problem a solution that may help.
2) If you are stuck go back through your work to see if something sparks your memory to continue.
3) Finally, do not be afraid to continue and be wrong, just learn from your mistakes and try another method, but first figure out why it was wrong so you do not make the same mistake twice.
Ryan Warner
1) Failing isn’t always the worst thing, you can learn a lot sometimes by failing
2) There is almost always more than one way to solve a math problem, when solving equations this is definitely true so try different ways if you can’t figure it out right away.
3) Learn everything you can about the equation first before you try to solve it.
Shannon Penix
1) It is important to look at the problem all the way through before trying to solve it, noticing things you already know.
2) It is alright to take a while to get through a problem and even not succeed. At least there is learning involved.
3) Take the time to go back over your work if you get stuck. There may be earlier steps that could trigger your thinking to continue.
Jeremy Sheaffer
1) Be fair to both sides of the equation.
2) Check your work
3) The harder something is to learn, the more chance there is that you will remember it.
Image credit: dullhunk @ Flickr
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