Why Is It Crucial to Check Your Work in Differential Equations?

[mathwonk]

I am grading differential equations finals.

In this subject almost every single answer can be checked by plugging it back into the equation to see if it works.

I even instruced students to "check your worK" on the final.

So far out of 112 problems graded, only one answer has been checked, and accuracy is running only about 40%.

Don't be that person who gets a 60 instead of a 90 because they refused to check an answer, to learn that it needed correcting.

i.e. don't be so foolish as to claim that y = x^2 is a solution of y''-2y'+y = x^2, just because that's what you got when you went through the procedure. try it and see. then do it again.

not checking your answer means you think you are perfect. Is that likely?

 

[leright]

I am grading differential equations finals.

In this subject almost every single answer can be checked by plugging it back into the equation to see if it works.

I even instruced students to "check your worK" on the final.

So far out of 112 problems graded, only one answer has been checked, and accuracy is running only about 40%.

Don't be that person who gets a 60 instead of a 90 because they refused to check an answer, to learn that it needed correcting.

i.e. don't be so foolish as to claim that y = x^2 is a solution of y''-2y'+y = x^2, just because that's what you got when you went through the procedure. try it and see. then do it again.

not checking your answer means you think you are perfect. Is that likely?

many times there is no time to check all of your answers though.

 

[chroot]

I'm glad you were never my professor, mathwonk.

- Warren

 

[kdinser]

many times there is no time to check all of your answers though.

This is so true of non math majors.

I believe that math classes beyond high school algebra need to be split into 2 divisions, math majors and everyone else. The everyone else group should still cover the same material that is covered by math majors and in the same way.

The difference would largely be in the problems given. First off, understanding that these students are not math majors and usually can't just whip this stuff off at lightning speed, a few less problems per test should be given, or the questions should be more routine. IMHO, it's insane to expect an engineering major to memorize every trig identity they are exposed to from trig to multivariable calc. When I took multivariable calculus, I spent way to much time before each test rememorizing obscure trig identities, integration tables, hyperbolic stuff, and derivative tables. Yes, I know most can all be worked out manually if needed, but there usually isn't time for that. This is the kind of stuff that is easily at ones fingertips should it ever be needed in the real world. Take this question for example:

dy/dx=(1/cosh x) find the general solution to this differential equation.

How does that question evaluate if I have learned how to solve separable differential equations better then:

dy/dx=2x*cos x^2 find the general solution to this diff eq.

The only difference is that one requires the student to have memorized something that most students will not need to have memorized and hardly ever run across in their classes, the other is a simple u substitution that anyone that completed clac 2 should be able to do. It would make a lot more sense to just force the student to learn the proofs of these methods, at least that would show a bit of understanding of the material.

Maybe I look at math the wrong way, but when the teacher is going through the proofs, I'm very attentive and awake, when they start going through specific examples, I can't stay awake at all because by that time, it's usually a memorization game.

 

[shmoe]

many times there is no time to check all of your answers though.

Of course there's some common sense involved in using your time wisely. For d.e.'s like the one mathwonk posted the time it takes to check should be a fraction of the time it took to come up with that attempted solution and well worth it.

 

[mathwonk]

.chroot, i am sure you could teach me to appreciate physics and i would hope i could also shjare my love for math with you.

 

[mathwonk]

as a previous poster said, if it takes a fraction of the time to check that it takes to work a problem, and the accuracy rate is 40% without checking, does it make sense to check?

do the math.

 

[mathwonk]

of course the problem of time is paramount. that is why we usually give tests which take us only 1/4 to 1/7 the amount of time that we allow the students to work them.

but think about it, if you cannot work a test in 3.5 hours which the teacher can work in 20-30 minutes, could there possibly be something lacking in your preparation?

oh yes, and the problems were chosen from a list handed out as homework weeks before the test.

try to get on the train.

 

[Maxwell]

3.5 hours?!

We get 2 hours for a final. I believe that is how most of the schools my friends go to do it as well.

You really give your students 3.5 hours??

 

[mathwonk]

our university gives 3 hours. so i try to make my finals about 2.5 times as long as a 50 minute test.

hence anyone should have plenty of time. but the older i get, the less i view a test as a contest, and the more I view it as a learning experience, so i try to give my class every opportunity to succeeed.

i even help them on problems if they are stuck. and i allow them to stay until the next class comes in, which is after 30 minutes, so if anyone needs 3.5 hours i allow it.

the point is to teach the material and to motivate people to learn it. the days are over when i viewed it as a contest to display my virtuousity.

my students are mostly struggling, and i am trying to help them realize their goals.

 

[dav2008]

Another important "checking your work" procedure is checking your units. Dimensional analysis is a subject that seems to be lost on many people.

It ranges from people multiplying a force by a distance and somehow getting power, to people getting ost when you ask them "if a current of 2A runs for 2 minutes, how many electrons have passed?" to people trying to add liter-atmospheres and Joules.

 

[chroot]

I don't know mathwonk, it seems to me (and always has) that you focus way too much on grading and ranking your students.

- Warren

 

[kdinser]

doh, can't get it working.

 

[mathwonk]

i agree chroot. but they force me to give grades here at the U. my preference would be a system where people get whatever they can out of the course and that is their reward, and i never have to waste my time giving them grades. I do not have such an ideal job though. But that is why I have volunteered for teaching in high schools and other situations where my work was unpaid and hence I could set my own agenda, not including ranking people.

In fact that is why I spend so much time here. let's see this is post number 2763, freely given advice and help, no compensation, and no grades or rankings.

I appreciate the chance to do this. Answering questions for those who merely want to learn is what keeps us going in our profession.

best wishes

 

[kudos213]

I think there is some importance and benefit to students receiving a grade for a class. Why go to college (as opposed to high school which is for the most part mandatory) then? Why not just check out textbooks and self-study or sit in on lectures but don't take exams. It's because traditional schooling offers discipline and structure and grades are a part of that. They reinforce successful work habits and help students understand what works and what doesn't. Perhaps the previous poster was saying that there was too much emphasis on grading but I do not agree that the exact opposite is the solution. However, this is my opinion...

In reference to checking your work...you do realize that if a student doesn't check their work they also don't have to see that their solution is wrong. If I'm confident I worked the problem correctly I'm more inclined to check it because I have confidence in my ability to solve the problem. In the end since you are the professor I think you can learn a lesson from your students as well, perhaps they are telling you something with their performance on these exams. In the end we must all learn...

 

[mathwonk]

i agree i can learn from my students. what i am learning is often that they are not like me, they do not seem to care to do well. they do not follow my advice on what will help them succeed. so this lesson in part helps me learn how to teach such students.

But it also gives me pause about what i should be doing for them. some of my students seem to want me to take more responsibility for their learning by using more tricks to enforce their practicing the skills they need.

I have always felt however that learning to take responsibility for ones own learning is an important part of growing up.

so I sacrifice some level of performnace on short term goals by my students, in favor of letting them grow into the adults they must be if they expect to excel at some point in the future.

when my own children were in high school, i let them make some mistakes i could have prevented because i knew they would soon be on their own in college and needed to learn the consequences of making those mistakes.

also for my students i sometimes choose to let them fail when they do not follow my advice, rather than force them to do so, when it will only mean they will lapse again when they are out of class.

some other stricter teachers have students who peform better on this semesters tests, but i hope mine will become self motivated learners sooner.

this is always an ethical problem for teachers: we know what will help our students, but should we force them to do it, or merely guide them to it, and let them discover the penalty for not doing so?

 

[mathwonk]

that is one of the reasons i like this forum: although I think I know what I am talking about, the anonymity of the forum allows even the most naive student to argue with me at will. so in the end only the persusaiveness of the argument matters, not the power relationship of teacher and student.

 

[Beeza]

I've always preferred the "hard-ass" professors that work us hard and don't accommodate for the immature students that like to spend their time drinking instead of studying.

 

[Markjdb]

Another important "checking your work" procedure is checking your units. Dimensional analysis is a subject that seems to be lost on many people.

It ranges from people multiplying a force by a distance and somehow getting power, to people getting ost when you ask them "if a current of 2A runs for 2 minutes, how many electrons have passed?" to people trying to add liter-atmospheres and Joules.

THANK YOU

It always annoys me as a student when I watch people spending all their time memorizing dozens of simple high school chemistry and physics formulas when they can just simply use the units to derive most of them. It's a lot easier, and it helps to make sure you're on the right track when solving some of the more involved (stoichiometry comes to mind) problems.

 

[Curious3141]

THANK YOU

It always annoys me as a student when I watch people spending all their time memorizing dozens of simple high school chemistry and physics formulas when they can just simply use the units to derive most of them. It's a lot easier, and it helps to make sure you're on the right track when solving some of the more involved (stoichiometry comes to mind) problems.

I found it most irritating that in my schooldays, the GCE papers were accompanied not only by log tables (which were OK for those who didn't have calcs), but also by formula sheets.

Now, I can't for the life of me imagine someone sitting for a calculus exam without being able to remember the quadratic formula (which was one of the things included on the sheet). Or the double angle trig formulae for that matter. The only thing I had difficulty remembering was the Factor Formulae, so I didn't bother - I just derived them on the spot with the compound angle formulae. For some reason, I trusted my own quick derivation more than the stupid formula sheet (which did have the FF).

The only thing those sheets are good for is physical constants, where it is better to set standardized values for all students to use. I know that if I had used my memorised values like c = 2.99792458*10^8 m/s or Avogadro's number = 6.022045*10^23 which were the best estimates of the time), my answers would've differed from the accepted answer keys used by the examiners.

 

[moose]

I still can't seem to understand some students in my physics course... They ask my help for something and I watch them try to solve some problem. They ask me if they got it right, their answer is usually something completely absurd which is obviously wrong. For example, they ask me if the answer is negative 4.35 seconds from time of whatever, or They say it takes .23 seconds for a traditional satellite to orbit the earth. Do they not think AT ALL?

Also, I have a friend who asks me what the formula is for *everything*. He is the type of person who would ask me what the formula was for getting the average of two numbers, or something else which a normal person wouldn't ever even think of remembering a formula for because it's just 100% obvious to anyone. Sometimes, he asks me if some formula is correct, even though it's obviously wrong, like he asks me if it's kq1r/q2^2... *blank stare*... The thing is, he isn't that stupid if he isn't caught up trying to think of a formula without thinking.

More on topic though, I don't really check my answers in calculus and physics and I do fine... I understand that this isn't differential equations or anything, but whatever.

 

[Pseudo Statistic]

Or the double angle trig formulae for that matter. The only thing I had difficulty remembering was the Factor Formulae, ...

Well, I've never heard of this "double angle trig formulae" or "Factor Formulae". :\ (I'm in Calculus right now... but never encountered these. Hm.)

 

[cogito²]

not checking your answer means you think you are perfect. Is that likely?

Well I actually am perfect, but that's because I do check my answers. ;)

 

[Markjdb]

Well, I've never heard of this "double angle trig formulae" or "Factor Formulae". :\ (I'm in Calculus right now... but never encountered these. Hm.)

The factor formulae (I believe) are formulas that allow expressions like
Ax^2 + Bx + C to be factored into their binomial factors. They would also include simple identities like x^2 + 2xy + y^2 = (x+y)^2 or x^2 - y^2 = (x-y)(x+y). There are many others, most a bit more complicated, but the factor formulae basically make it easier to factor certain expressions rather than using a "guess and check" method to find the factors.

The double angle trig formulae are the following identities:
sin(2x) = 2*sin(x)*cos(x)
cos(2x) = 1 - 2*sin^2(x)
tan(2x) = (2*tan(x))/1-tan^2(x)

 

[Curious3141]

The factor formulae (I believe) are formulas that allow expressions like
Ax^2 + Bx + C to be factored into their binomial factors. They would also include simple identities like x^2 + 2xy + y^2 = (x+y)^2 or x^2 - y^2 = (x-y)(x+y). There are many others, most a bit more complicated, but the factor formulae basically make it easier to factor certain expressions rather than using a "guess and check" method to find the factors.

The double angle trig formulae are the following identities:
sin(2x) = 2*sin(x)*cos(x)
cos(2x) = 1 - 2*sin^2(x)
tan(2x) = (2*tan(x))/1-tan^2(x)

No, Factor Formulae allow one to convert from Products of trigonometric ratios to sums (or differences) and vice versa. Also known as Product to Sum formulae. http://library.thinkquest.org/C0110248/trigonometry/formfactor.htm

I didn't remember them because I could derive them easily by seeing that cos(x+y) + cos(x-y) = 2cosxcosy and so forth.

 

[Hootenanny]

Okay here's one for you (from personal experience). You have ten mintues remaining on your test. You have just finnished the paper, but have left one question out as you were having difficulty doing it. The question is worth a significant number of marks, let's say about 4% of the exam total. Do you attempt this question with the remaining time; or do you check you other answers?

~H

 

[bluechina]

i'd better take ur advice i think ,'cause i always did a bad job without checking my answers during the examination .though i got enough time to do that

 

[exequor]

In engineering I believe in the application of math and not just memorizing a few applications such as mixing problems, springs, etc. Afterall one day you will come across and application that cannot be solved by one of those typical models, then what?

I think most math exams are speed tests, some professors even claim that given time most students could solve the problems.

 

[Curious3141]

Okay here's one for you (from personal experience). You have ten mintues remaining on your test. You have just finnished the paper, but have left one question out as you were having difficulty doing it. The question is worth a significant number of marks, let's say about 4% of the exam total. Do you attempt this question with the remaining time; or do you check you other answers?

~H

Of course you do the last question. By just showing working, you're guaranteed at least 20 to 30 % of the total mark for the question in most school level exams. Unless you're the sort who's prone to making tons of careless mistakes (unfortunately, I'm this sort ), you should trust that most of the others would've come out perfectly. In any case, even with minor errors in the final answer, if the working is completely correct conceptually for the rest, you'd get at least 80% of the mark, so checking will only add 20% at most.

 

[kdinser]

Of course you do the last question. By just showing working, you're guaranteed at least 20 to 30 % of the total mark for the question in most school level exams. Unless you're the sort who's prone to making tons of careless mistakes (unfortunately, I'm this sort ), you should trust that most of the others would've come out perfectly. In any case, even with minor errors in the final answer, if the working is completely correct conceptually for the rest, you'd get at least 80% of the mark, so checking will only add 20% at most.

This is totally at that the whim of the professor. I've had some teachers that gave 50% just for writing down the correct formula or drawing a picture of the problem. I have also had professors that immediately took off 50% for a wrong answer and then back tracked and took more off for each mistake that led up to the wrong answer and any missing steps.

Given these 2 examples, in the first case, you would be better off working on the skipped problem. In the second case, your time would be best spent looking over the problems you were able to do. I guess you have to know your teachers grading style.

 

[Brad Barker]

some professors don't really accommodate work-checking. my intro to theoretical phys professor also taught elementary differential equations and the calcs, in various years. his approach to testing was to make the test "unfinishable." most students weren't able to complete the exam or really check their answers. i'd be able to finish, but going at full-speed and without really being able to check. (my personal opinion is that this isn't very fair to the students.)


another anecdote: i got a 70-something instead of a 90-something on a four question relativity exam because i didn't read through the instructions all the way. i saw a problem that looked familiar, so i worked out what i thought the question asked and not what the question ACTUALLY asked.

so when you check your work, i guess, also take the time to re-read the instructions!

 

[mathwonk]

that is good advice about rereading the questions.

As to the time needed to finish an exam, i usually give over an hour for a test that i can finish in 8-12 minutes, and many people are still "working", i.e. scratching their heads, at the end. so how long it takes often depends on how well one knows the material.

e.g. to solve the de: y''+y'+y = 2x-x^2, note that the LHS is (1+D+D^2)y, which has inverse (1-D) modulo monomials of degree at most 2, i.e.

(1+D+D^2) (1-D)f = (1-D^3)f = f, if f is a polynomial of degree 2 or less, since D^3 f = 0 for such f.

so the answer is y = (1-D)(2x-x^2) = 2x-x^2 -2 +2x = -2+4x-x^2, which takes only a few seconds.

check: -2+4x-x^2 + 4-2x -2 = 2x-x^2. ok.

of course if one refuses to learn about operators and uses a more cumbersome method, like variation of parameters, it takes longer.

 

[amenon]

I completely agree that checking your work will do you good. But that means going back and doing the test again. That takes a long time.

 

[mathwonk]

the point is to practice taking tests before going into the test room. time yourself, and work at completing a test with checking, in less than the time alotted.

how many times have you prepared for a test by writing a sample test, longer than the one you expect, then taking it under timed conditions, and checking it?

never? what kind of student are you? you can do as well in school as you wish to do, it all depends on how much effort and time you want to expend at it.

 

[Manchot]

and many people are still "working", i.e. scratching their heads, at the end. so how long it takes often depends on how well one knows the material.

You shouldn't judge your students negatively if they take the entire period to take an exam. I almost never leave early when I'm taking an exam, though I'm almost always done early. The way I see it, if I have the time to go back and check two, three, even four times, there's no harm done in doing so.