and Grading Partial Credit
For starters, here is some simple advice.
MTH 103-095
You would be amazed at the number of times students seem to make mistakes because they read their own writing incorrectly. You should write all graded items in PENCIL. You are much more likely to solve problems correctly if you organize your steps and write legibly. Of course, this is much easier if you can erase mistakes.
Here is a detailed explanation of my grading philosophy. Much of this is probably shared by every math instructor. For almost every problem, grading is based on the following things (in the order indicated).
The first and most important mission students must accomplish in their response to every problem is to demonstrate that they understand the purpose (goal) of the problem. Most students usually do this successfully. However, the most common way students fail to do this is to write the wrong TYPE of answer in the answer blank. For example, suppose a problem asks students to find the solutions of an equation. Solutions of an equation are numbers. So if a student writes numbers in the answer blank, that's good. If they are the correct numbers, then that's even better, but this is of much less importance. If students write something other than numbers in the answer box, then they must not have read the question to find out what the purpose (goal) of the problem was; or even worse, perhaps they read the question but did not know what the word "solution" meant. This is why it is crucial to learn the vocabulary associated with the course. The most catastrophic error a student can ever make is to give a final answer that cannot possibly be correct because it is not even the type of answer the problem asked for. NEVER write an answer on the answer blank that you know cannot possibly be correct. To avoid such catastrophic errors, try some of the following strategies. These are especially helpful when taking tests.
o
After you read a problem and before you do anything else, decide what type of object the answer is. Is it a number? A set? An expression? An equation? An inequality? A function? A matrix? A vector? A scalar? What type of object is it? You can tell this ahead of time in almost every problem you will ever face.
o Begin every story problem by "answering the question before you have answered the question." Read the directions, then go immediately to the answer blank and write a complete sentence that will eventually serve as the final answer to the problem. Omit only the numerical portion of the answer for now. You can plug it in once you get it.
o
What variables appear in the given problem? Which of those variables will appear in the final answer? You can usually determine this before you actually find the answer, and by thinking about this ahead of time, you can catch many catastrophic errors. If you know ahead of time that the answer should depend on the value of h, yet h does not appear in your final answer, then something must be wrong!
o If you can tell that something must be wrong, SAY SO! Sentences like "there must be something wrong, because this result shouldn't be negative" show that you know what you are talking about, but perhaps messed up something in the details. Using comments like this on your paper is good "damage control" for problems you mess up.
Grading and Partial Credit
MTH 103-095
The second most important part of a student's response is the method. There is usually more than one possible method that can be used to solve each problem. Did you identify one of the possible methods? Is the strategy you chose appropriate? Did you recognize which concepts are applicable to the problem? If so, your method is good. As a non-example of identifying a valid method or appropriate strategy, suppose that an algebra test asks students to find solutions of the equation x 3 - 2 x - 4 = 0 , and a student tries to use the quadratic formula. This is a grievous error, since it indicates that the student either failed to recognize that the equation was not quadratic, or perhaps did not realize that the quadratic formula is only applicable to quadratic equations (hence the name "quadratic" formula). To avoid errors like this, you must study not only methods for solving various problems, but also when those methods are applicable. The third most important part of your response is your solution. Assuming you identified a valid method and general strategy to use for the problem, did you correctly apply that method? Is your work written correctly, logically, and clearly? For many beginners, this one is the most elusive of the four parts of the response. For that reason, a detailed discussion of how to write a good solution to a math problem is given on the next few pages. Read on. One of the most common types of solution that beginning math students must write is for the following type of problem: You are given an object, and you are asked re-express it in a different form--that is, write down another representation of the same object. Factoring or expanding polynomials, simplifying or evaluating expressions, and doing basic arithmetic are examples of this situation. For example, consider the following two problems. #1. Simplify (8 + 24 ) / 2 . #2. Expand ( x + 4)( x 2 + 3 x) . (The answer is 4 + 6 .) (The answer is x 3 + 7 x 2 + 12 x .)
At first glance, these two problems may appear to have nothing in common. But they do. In both cases, the final answer is exactly equal to the given object. Here's an explanation.
Consider problem #1 above, for instance. The number 4 + 6 is exactly the same as the number (8 + 24 ) / 2 . If you are not fully convinced that this is indeed the case, you may want to check for yourself by using a computer to see rounded-off decimal expansions of (8 + 24 ) / 2 and 4 + 6 . You will see that their decimal expansions are the same (they begin with 6.4494897427831780981972840747059 but do not stop, since this number is irrational). Therefore, 4 + 6 is just a different representation of (8 + 24 ) / 2 . Do not think of 4 + 6 and (8 + 24 ) / 2 as different numbers--they are not different numbers! Instead, realize that 4 + 6 and (8 + 24 ) / 2 are different representations of the same number. The directions for problem #2 said "simplify". To "simplify" something does not mean to change its "identity" or "meaning"; but rather, to simplify something means to change only its form so that the new form is more concise, clear, or compact. So, for any problem that says to "simplify" some object, the first thing to know is that the final answer must be exactly equal to the given object.
Grading and Partial Credit
MTH 103-095
Now consider #2, which said "expand ( x + 4)( x 2 + 3 x) ". The direction "expand" can be thought of as a more specific version of the direction "simplify". To "expand" an object (which, in this case, is an algebraic expression) means to rewrite the given expression so that it contains terms (things that are added), not factors (things that are multiplied). In the sentence you just read, the important word was REWRITE. Go back to that sentence (which began with "To `expand' an object"), read it again, and this time emphasize the word "rewrite" as you read it. Again, the directions indicate that the goal is NOT to change the given object, but rather, to write down another representation of the same object. In other words, the goal of the problem is to write an expression that is exactly equal to the given expression, and more specifically, the "new" expression should have terms instead of factors. Indeed, the final answer x 3 + 7 x 2 + 12 x is exactly equal to the given expression ( x + 4)( x 2 + 3 x) . The reason these two expressions are "equal" is that no matter what value we substitute into the expressions for x, the values of the expressions are the same. For example, if we substitute (say) - 1 for x, we see that the value of the given expression x 3 + 7 x 2 + 12 x is (-1 + 4)((-1) 2 + 3(-1)) = (3)(1 - 3) = (3)(-2) = -6 and the value of the final answer expression, ( x + 4)( x 2 + 3 x) , is (-1) 3 + 7(-1) 2 + 12(-1) = -1 + 7 - 12 = -6 . So, both expressions' values were - 6 when we substituted in - 1 for x. Instead of - 1 , if we were to substitute (let's say) 2 for x, we would find that the expressions' values are (2 + 4)(2 2 + 3(2)) = (6)(4 + 6) = (6)(10) = 60 and (2) 3 + 7(2) 2 + 12(2) = 8 + 28 + 24 = 60 , which again are the same. As these "for instances" suggest, the values of the expressions ( x + 4)( x 2 + 3 x) and x 3 + 7 x 2 + 12 x are the same, no matter what value we substitute for x. Therefore the given expression ( x + 4)( x 2 + 3 x) and the final answer x 3 + 7 x 2 + 12 x are exactly equal, and again, the final answer is exactly equal to the given object.
Let us summarize the explanations for why problems #1 and #2 have very similar goals. The common feature was that in both problems, the goal was to find a different representation of the given object. Therefore, the final answers were exactly equal to the given objects. It is standard practice to write solutions for problems like this in the following format, or a similar format. [Given Object] = [Another representation of the given object which is in a form "closer to" the desired form] = [Yet another repre]
Grading and Partial Credit
= [Same object, intermediate form #n] = [Same object, desired form #1]
MTH 103-095
You can open any math book and see the above format being used. The only variation from one type of problem to the next is "desired form" is (i.e., the appearance of the final answer). This is always made clear by the directions, or the context of the problem.
Grading and Partial Credit
MTH 103-095
Here is another sample problem with a few different example solutions, all of which use the same method and arrive at the same answer, but have differently-written solutions. Problem Factor 2 x 3 - 8a 2 x + 24 x 2 + 72 x into prime polynomials with integer coefficients. Solution (one acceptable format) 2 x 3 - 8a 2 x + 24 x 2 + 72 x = 2 x 3 + 24 x 2 + 72 x - 8a 2 x = 2 x[ x 2 + 12 x + 36 - 4a 2 ] = 2 x[( x + 6) 2 - 4a 2 ] = 2 x( x + 6 + 2a )( x + 6 - 2a ) Solution (another acceptable format) 2 x 3 - 8a 2 x + 24 x 2 + 72 x = 2 x 3 + 24 x 2 + 72 x - 8a 2 x = 2 x[ x 2 + 12 x + 36 - 4a 2 ] = 2 x[( x + 6) 2 - 4a 2 ] = 2 x( x + 6 + 2a )( x + 6 - 2a )
Solution (still another acceptable format) 2 x 3 - 8a 2 x + 24 x 2 + 72 x = 2 x 3 + 24 x 2 + 72 x - 8a 2 x = 2 x[ x 2 + 12 x + 36 - 4a 2 ] = 2 x[( x + 6) 2 - 4a 2 ] = 2 x( x + 6 + 2a)( x + 6 - 2a) The three versions of the same solution all have two very important commonalities: The object written at each step was EXACTLY EQUAL to the entire object written at the previous step. Equality was never lost. The solutions are complete English sentences which make complete sense when read out loud. For example, to read any of the three above solutions, you could say this: " 2 x 3 - 8a 2 x + 24 x 2 + 72 x equals 2 x 3 + 24 x 2 + 72 x - 8a 2 x , which is equal to 2 x[ x 2 + 12 x + 36 - 4a 2 ] , which equals 2 x[( x + 6) 2 - 4a 2 ] , which is equal to 2 x( x + 6 + 2a)( x + 6 - 2a) ."
Grading and Partial Credit
MTH 103-095
As a non-example of how to write a solution to a problem in which the goal is to give another representation of the same object, the following is an unacceptably poor solution of the same problem. Poor solution 2 x 3 - 8a 2 x + 24 x 2 + 72 x 2 x 3 + 24 x 2 + 72 x - 8a 2 x 2 x[ x 2 + 12 x + 36 - 4a 2 ] ( x + 6) 2 ( x + 6 + 2a )( x + 6 - 2a ) = 2 x( x + 6 + 2a )( x + 6 - 2a ) The above solution is poor for the following reasons:
Equality was lost. The objects on the fourth, fifth, and sixth lines are obviously not equal to each other. To make matters worse, the author claimed that the object on the fifth (second-to-last) line is equal to the object on the last line. That is, the writer said that ( x + 6 + 2a )( x + 6 - 2a ) is exactly equal to 2 x( x + 6 + 2a)( x + 6 - 2a) ; but this is false. There is only one assertion (i.e., statement) in the entire solution, the namely assertion that ( x + 6 + 2a )( x + 6 - 2a ) = 2 x( x + 6 + 2a)( x + 6 - 2a) ; and this assertion is false as already mentioned. Aside from that, there are no other statements in the solution; there is only a list of objects. The reader cannot tell what the author is saying about the objects 2 x 3 - 8a 2 x + 24 x 2 + 72 x and 2 x 3 + 24 x 2 + 72 x - 8a 2 x . Did the author mean to indicate that these expressions are equal? Did the author actually know they are equal? Further, does the author even understand what it means for two algebraic expressions to be equal? The reader cannot tell. In particular, a grader who reads this solution would probably doubt that the author understands the following concept: Algebraic expressions are equal if and only if the value of the expressions are the same for any values that are substituted for the variables in the expressions (provided the values of the variables are in the domains of the expressions).
So the reason why 2 x 3 - 8a 2 x + 24 x 2 + 72 x and 2 x 3 + 24 x 2 + 72 x - 8a 2 x are equal is that if we to substitute (let's say) 5.76 for x and - 1.8 for a into both expressions (or any other values of x and a that we pick), the values of the two expressions would be the same. If the solution's author had understood this, then presumably the author would have indicated his or her understanding by actually saying in the solution that 2 x 3 - 8a 2 x + 24 x 2 + 72 x and 2 x 3 + 24 x 2 + 72 x - 8a 2 x are equal. Furthermore, if the author understood what it means for algebraic expressions to be equal, then he or she would certainly not have written down the false assertion that ( x + 6 + 2a )( x + 6 - 2a ) equals 2 x( x + 6 + 2a)( x + 6 - 2a) . In summary, the poor solution on the previous page shows either a lack of mathematical understanding or a lack of effort (or both) on the part of the author; and therefore the poor solution certainly should not earn the author full credit.
Grading and Partial Credit
MTH 103-095
In contrast to the previous example, it is not always correct to write down the phrase "is equal to" (i.e., the symbol "=") between steps. Here is an example. Problem Solve the equation 2 x - 8 = 7 - 3 x . Poor response
2 x - 8 = 7 - 3x = 5x - 8 = 7 = 5 x = 15 = x=3
This time the author has claimed that for each of the objects 2 x - 8 , 7 - 3 x , 5 x - 8 , 7, 5 x , 15, x, 3 is equal to ALL the other objects. For example, the author has stated that 7 is equal to 15 and also equal to 3. The author also said that 2 x - 8 equals 5 x - 8 , and 5 x is equal to both of those; and so on. This is ridiculous. Had the author understood that the symbol "=" means "is equal to", and understood what it means for expressions to be equal, he or she would not have written this response. Here is a response to the problem that makes sense. An acceptable response 2 x - 8 = 7 - 3x 5x - 8 = 7 5 x = 15 x=3 The format in the above response is how tradition says we search for solutions of equations in one variable--we write down a list of equations that get simpler and simpler until we get one whose solutions are obvious. But the traditional approach is not always the most logical or best way to do things (this holds true for life in general). Below are two more responses that are (in a sense) a bit more logical. When you read the next two responses to yourself, read " " by saying "implies that" or "means that" or "which implies that", etc. Another good response 2 x - 8 = 7 - 3x 5x - 8 = 7 5 x = 15 x=3 Still another good response 2 x - 8 = 7 - 3x 5x - 8 = 7 5 x = 15 x=3 Notice that both these responses are read out loud in the same way, and both make sense when spoken in English.
Grading and Partial Credit
MTH 103-095
Recall that the most important part of your response to a math problem is to show that you understand the goal or purpose of the problem, and after that, the second most important part of your response is to choose a method that is appropriate to accomplish the goal, and the third most important part is to write a good solution. The solution is the aspect of the response that this lengthy discussion (which astonishingly began all the way back on page 2) has addressed. You may have noticed that the discussion of the method and the purpose parts were considerably shorter than the discussion of the solution. This is because beginners usually struggle the most with the solution part, perhaps due to the fact that in students' prior math courses, the final answer was what mattered most, and so little if any focus was placed on communicating the mathematics clearly and correctly. But here are some points of encouragement, which I hope will be pleasant reading for all serious students. First, there is no need whatsoever to memorize "the one correct" format for solving each type of problem faced. This is because no problem has only one correct strategy for solving it nor a single correct format for presenting the solution. Just as there are many legitimate ways to say the same thing in English, there is more than one way to write a perfectly good solution to a math problem. In fact, (here comes the second point of encouragement) the criteria for writing good mathematical solutions are the same as the familiar criteria for writing good English statements--in either case, the writing is "good" if it makes sense when read aloud, if it is organized logically, if it addresses the points concisely, and so forth. So, the very same mindset applies both to writing in plain English and writing solutions to math problems. In both cases, you can simply proofread what you have written, and perhaps also have another person (with an appropriate level of experience) proofread it as well, so that you can verify your work is understandable. Of course, mathematics uses lots of symbols and abbreviations, but each symbol and each abbreviation represents an English word or phrase; so if you simply know how to read the symbols, they will present no additional difficulty to you as you proofread your writing. Now for the third and most important encouraging point: The effort you put into learning effective mathematical communication will pay you back great dividends, because a far greater understanding and appreciation of the mathematics is gained when you are able to explain the mathematics you are doing. You surely have your own personal experiences that convince you of this--whether in a mathematical setting or not, your understanding and appreciation of any subject tend to increase when you explain it to someone. This is why communicating math effectively will help you learn math better--a good solution for a math problem is (at least in part) an explanation that leads the reader logically from the problem definition to the result. Do not think of "the reader" as only "the grader". The first person who reads your solutions is always you! Thus, your solutions to math problems will function most effectively if you author them to serve as explanations (to yourself) of your own reasoning. In doing this, you will accomplish a goal far more important than pleasing a grader. You will also secure a more permanent understanding of the topic at hand, so that your mathematical ability ...