Grading math: How much would you take off for this mistake?

[EKS]

I'm trying to get a handle on my 16yo son's math issues at his private school.

 

On his most recent test, there is a problem with a fourth degree polynomial where the coefficients for the second, third, and fourth terms are given as a, b, and c, respectively. The student is supposed to figure out what a, b, and c are given two factors as well as a third expression that has a remainder.

 

So, if you're still following this (and if you are I thank you!), the student must perform polynomial division three times dealing with terms with coefficients like 2a+b-c. Then he must use what he gets as a quotient for each division problem (setting it equal to its remainder) and figure out what a, b, and c are by solving several sets of simultaneous equations.

 

So, we are finally getting to my question. Say the student made a single sign error in one of the quotients but that everything else was correct, both before and after the sign error. Keep in mind that this problem involved an entire page of calculations. The problem is worth 8 points. How much would you take off?

[Jane in NC]

Did the student perform the same level of complexity despite the sign error? I am tempted to say just one point given the amount of work involved with the problem. If, however, the sign error led to a problem of greater simplicity, I would deduct more.

 

Demonstrating the mechanics of problem solving is far more important to me here than the sign error.

[Arcadia]

For school, we got two points deducted for wrong answers and a point deducted for each careless mistake. So three points would be deducted for that question.

[regentrude]

Did the student perform the same level of complexity despite the sign error? I am tempted to say just one point given the amount of work involved with the problem. If, however, the sign error led to a problem of greater simplicity, I would deduct more.

 

Demonstrating the mechanics of problem solving is far more important to me here than the sign error.

 

:iagree:on all points.

[Candid]

I also had teachers who warned us that if we had continuous problems with careless errors penalties might become stiffer if we did not take action to reduce them. I think that is more common in high school than college.

Maybe I'm over influenced by the last thing I've read, but having just read Salman Khan's book about the Khan Academy, I'm trying to focus more on mastery of math topics with mine. He tells the story of his 12 year old niece, whom he started tutoring because she failed to get a high enough grade on some entrance test to get tracking into the highest math track in middle school. He knew she was a smart kid, even though she herself thought she was "dumb at math".

 

He quizzed her, and realized there was one math topic, units (and conversion thereof) , which she just didn't get. She had always gotten "A"s in math, but, he points out, you can get 95 percent on all your tests, and still have big holes in your background, holes which can be a big problem later in your math career, when they become needed foundation. He focused a lot of his one-on-one tutorial with her on unit conversion, she was able to re-take the test, passed with flying colors, and many years later, she considers herself good at math.

 

One of his solutions to this problem is to test to mastery. That is, you don't "pass" a test until you can completely correctly do ten problems in a row.

 

I know this doesn't answer your question, and the knee-jerk reaction is to simply case "concepts important, silly mistakes less so", but, ultimately, math is about getting the correct answer, and mistakes due to inattentiveness can happen even when using calculators and computers. Learning how to focus, and to double-check to correct for silly mistakes is an important part of learning math, as well as learning the concepts themselves.

[RootAnn]

If *I* was grading it, I'd take off two points.

 

However, I remember one test in college (Calc 3) where there were only four questions on the entire test. I did every single thing correct on one of the problems until the final step. At the very bottom of an entire page of calculations, I wrote:

 

= sq root of 49

= 9

 

I missed all 25 points for that problem. (This from a teacher who CONSTANTLY made silly mistakes like this on the board when he was working problems for us in class.)

 

So, it really does depend on the teacher & their policy. As long as they are consistent, there isn't really anything you can complain about, IMO.

[Cosmos]

I'm trying to get a handle on my 16yo son's math issues at his private school.

 

On his most recent test, there is a problem with a fourth degree polynomial where the coefficients for the second, third, and fourth terms are given as a, b, and c, respectively. The student is supposed to figure out what a, b, and c are given two factors as well as a third expression that has a remainder.

 

So, if you're still following this (and if you are I thank you!), the student must perform polynomial division three times dealing with terms with coefficients like 2a+b-c. Then he must use what he gets as a quotient for each division problem (setting it equal to its remainder) and figure out what a, b, and c are by solving several sets of simultaneous equations.

 

So, we are finally getting to my question. Say the student made a single sign error in one of the quotients but that everything else was correct, both before and after the sign error. Keep in mind that this problem involved an entire page of calculations. The problem is worth 8 points. How much would you take off?

 

Did he perhaps make the problem more complicated than it needed to be? Perhaps I'm misunderstanding the problem, but from your description it appears to me the simplest way to solve the problem would be to expand the factored terms and then match up the coefficients with a, b, and c. No polynomial division or simultaneous equations needed.

 

Again, perhaps I'm misunderstanding the problem. I mention the possibility because when I was teaching math I sometimes found that students would solve a problem using a far more circuitous path than necessary. The additional steps did often introduce more opportunities for error, unfortunately. But I did not adjust my grading scheme to account for a student's choice to solve it a "hard" way. Is that maybe what happened here?

[jennynd]

Depends on the test. If it is a quiz, I will take 4 points off. Just so the kid learn the lesson to be careful. In a real test. I will take 1 point off

[Jane in NC]

 

I know this doesn't answer your question, and the knee-jerk reaction is to simply case "concepts important, silly mistakes less so", but, ultimately, math is about getting the correct answer, and mistakes due to inattentiveness can happen even when using calculators and computers. Learning how to focus, and to double-check to correct for silly mistakes is an important part of learning math, as well as learning the concepts themselves.

 

My reaction of concepts over sign errors is far from "knee jerk". Years of teaching college level mathematics courses has led me to this conclusion.

 

Mathematics is not about "getting the correct answer". This may be partially true when teaching arithmetic, but mathematics in higher level classes is about process. After the basic Calculus sequence, mathematics is focused on proof. There are no answers per se; there is only methodology.

 

Trust me--sign errors are not encouraged but are all too common. When a student has done a page or two of calculations, a heavy penalty for a sign error serves no point in my estimation.

[KarenC]

I would take off 1 point. It's so easy to lose focus in a long, complicated problem.

 

Karen

[regentrude]

Mathematics is not about "getting the correct answer". This may be partially true when teaching arithmetic, but mathematics in higher level classes is about process. After the basic Calculus sequence, mathematics is focused on proof. There are no answers per se; there is only methodology.

 

Yes.

Which, in turn, also means that a student who arrived at a "correct" answer by means of a faulty reasoning process may find himself receiving hardly any credit for the problem, despite having the right expression as his final result.

My reaction of concepts over sign errors is far from "knee jerk". Years of teaching college level mathematics courses has led me to this conclusion.

 

Mathematics is not about "getting the correct answer".

 

First off, the OP is talking about high school math, and even in a lot of higher level math, getting the correct answer is key. If a bridge or financial firm fails because of a math error, no one says "well, at least it was a sign error, and not a conceptual error".

 

Moreover, this really gets to the idea of assessment. What is "one point" that everyone wants to take off, no doubt trying to preserve a higher grade? What does an "A" on a given test mean? If a test has 10 such problems, each of which has a careless error, is that still an "A" test? I'm not interested in assuring that my kids get good grades, I want them to learn math, and to prove to me that they have, and have learned it in a way that they've retained it. Too many school kids are expert test-takers, and really understand little of the topics at hand.

[regentrude]

Moreover, this really gets to the idea of assessment. What is "one point" that everyone wants to take off, no doubt trying to preserve a higher grade?

 

One point would correspond to a deduction of 12.5% for the problem in question (there were 8 total points); this would mean the student has performed at B level on this problem. If everything else is correct, this level of mistake in a problem as described would correspond to a B performance, not a C performance.

 

I am a college instructor and give assessments to test whether my students have mastered material. My tests are designed to test a variety of concepts, and different mistakes are weighted to different degrees. A mistake that shows a lack of basic understanding is penalized by a higher deduction than a simple arithmetic mistake. This is not to preserve a higher grade, but because the arithmetic mistake with correct conceptual work demonstrates a higher degree of knowledge than the conceptual error. Every one of my colleagues grades this way.

It is not realistic to expect even the strongest students to perform completely without errors (in fact, if too many students get everything right, that means the test was too easy). Otherwise, you would not need to assign several points to a longer problem - you could have an all multiple choice format for correct answer only., I don't know of any math course run this way, because it is not a sensible assessment of the student's capabilities.

 

What does an "A" on a given test mean?

In my classes, an A means that the student has displayed an exceptional degree of mastery, which about 15% of the students achieve; the average is a C.

Giving varying degrees of partial credit for varying levels of mastery allows me to differentiate between students who know the material somewhat, who know it well, and who excel.

Edited November 5, 2012 by regentrude

[kiana]

In my personal opinion, the purpose of marking a test is to determine what proportion of the course subject a student knows. A student who knows more should receive a higher grade than a student who knows less. Assigning the same problem grade to a student who works nearly everything correctly but makes a computational error as to a student who simply writes 'idk' and moves on makes my test not very useful.

 

With careless errors, I usually take off one point for the first couple and two after that. This has the effect that a student who frequently makes careless errors but does understand the concepts cannot make more than a C. I may take off more than that if the problem is clearly wrong in a "you should have known that before taking the course manner" -- i.e. probability greater than 100%, negative length in a word problem, etc.

 

Now, in a homeschool environment, you can stop on a topic and say 'we'll move on when you can do this perfectly.' In a school environment (which is what the OP was posting about), that's not a realistic idea, and frankly, neither is failing students who know about 80% of the course material.

[EKS]

Thank you all for your input on this. I really appreciate it!