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Chalk Dust Company
PMB 256
16107 Kensington Dr
Sugar Land, TX 77479-4401

800.588.7564 (USA)
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Evaluation and Assigning Grades

Next to choosing a curriculum, making out tests and assigning grades is likely the most daunting task for homeschooling parents. That is understandable since the matter of "tests and measurements" represents an entire semester course of college study for secondary education majors.

The basic concepts involved in that course are presented below along with specific adjustments and suggestions for homeschooling a math curriculum. The discussion is divided into two parts. An outline form is used for easy reference.

Part I, entitled Tests, discusses techniques for making out tests and quizzes and Part II, entitled Measurement, addresses methods for grading tests as well as assigning course grades.

Part I - Tests

Some parents, perhaps most, will choose to use the tests printed in the textbooks for evaluation purposes because parents simply don't have the time, much less the experience, to make out a test for each chapter. That is expected, but occasionally students need to be re- tested and without a second printed test parents are left to make out their own test. The following is a guideline to assist in that process.

A. General factors to consider in making out a test:

1. Fairness
  a. Instructions are clearly stated
  b. Expected performance is understood
  c. Method of grading is understood
  d. Value of each problem is understood
  e. Level of difficulty is not beyond homework
     (except for extra credit problems)
2. Depth - all major concepts within a chapter are tested
3. Breadth - various levels of difficulty within each topic are tested

B. General Comment

Allow the student as much time as needed. Test-taking is an important part of the overall learning experience because students learn to think clearly and carefully when allowed and encouraged to take their time. Some say that timed tests teach students to work quickly. In my view, that practice leads to anxiety, frustration, a general sense of unfairness, and fundamentally undercuts the goal of teaching organizational skills and logical thinking.

C. Test Item Selection

Choosing problems which cover both the depth and breadth of content within a chapter sounds logical and easy on the surface. But once problems begin to be chosen you soon realize that there is a tendency to choose too many problems, particularly in long chapters. The solution is to choose problems from the review exercises. Since students have (probably) worked the odd-numbered problems, you may wish to limit your choices to the even-numbered items.

[Actually, there is no harm in choosing problems which a student has already worked. Students are not likely to memorize answers and the method of solution is worth more than the answer anyway. More on that later.]

A brief investigation reveals that, in most textbooks, a list of even-numbered problems from the review exercises will still provide too many problems for a text. The selection may further be limited by selecting every other even problem (2, 6, 10, 14, 18, etc.). Now look at the problem set to be sure that depth and breadth issues are covered and, particularly in short sub-sections, you are not consistently choosing either the easiest or hardest questions.

Even though the time allowed for students to take a test should not be limited, the number of test questions should be based on the number which students are reasonably expected to complete in about an hour. Parents may wish to test more often than chapter intervals if they feel that the number of test items required for accurate evaluation may exceed that guideline. Notice that most textbooks contain mid-chapter quizzes and those may be used as a source of test questions when deciding to test at mid-chapter intervals.

Part II - Measurement

A. Concept

Briefly, the measurement process is one of assigning a grade to a course by evaluation of student performance gathered from a variety of sources. The sources may include homework, daily or section quizzes (announced or not), chapter tests, mid-term tests, and a final exam. I've used all of those sources in classrooms at various grade levels. Some are used as much (or more) for motivation as they are for performance evaluation. [Think about that a moment in terms of your own experience and your own child.]

Fortunately for parents, there are choices in the process and those choices may (and should) be made according to student response. That is, some students stay on task better with more frequent evaluation while others find frequent evaluation intimidating. Most students are grade conscious but that does not necessarily imply a need for frequent testing. Students who tend to forget mathematical concepts quickly should be tested frequently.

B. Comment on Forgetful Students

Lack of retention is not an unusual condition and it is quite often age related; that is, it goes away remarkably quickly once the child reaches some elusive and unpredictable level of maturity. Don't be too concerned if your child can't remember how to solve a simple linear equation after knowing the concept just two weeks ago.

It is true that math is generally foundational in nature; that is, concepts in an early chapter will likely be used to some degree in later chapters, so retention is certainly desirable. But too much emphasis on knowing all previous material can actually contribute to the problem. The way to deal with it is to test frequently and when previous information is needed but forgotten, simply review it without making a big deal out of it.

On a larger scale, lack of retention is the main reason for using the spiral approach in teaching. If you are unfamiliar with the spiral approach, it involves the use of previously learned material to develop new material. An example in Algebra 2 is the topic of factoring. Factoring in Algebra 2 includes methods learned in Algebra 1 plus a couple of more advanced methods.

The spiral approach in teaching from one grade level to another is nothing more than review in a wide time frame, a time frame so wide that students are expected to forget. Why should we expect complete retention in tighter time frames. What is the magic time frame for complete retention? The answer is, it depends on the student, and it changes with age.

Let's get back to measurement.

C. Homework - Part 1

Giving credit for homework is optional. Homework can be regarded as a necessary part of preparation for tests and, therefore, not worthy of a grade. On the other hand, some students refuse to do anything unless they are rewarded. [I wonder how that started. Okay, it's a separate issue but one worth pondering in your spare time.] I'm not putting down the rewards program, mind you, because for some students it's a much easier motivator than trying to teach a philosophical work ethic.

If you decide to grade homework; or, more accurately, if you decide to give some credit for the completion of homework, then I suggest an awards program containing both positive and negative possibilities. More on that later.

First understand, though, that you are giving credit for the attempt, and not necessarily for success, because homework is a place where mistakes are expected and learned from. Homework is the mechanism for the transfer of information from short term memory to long term memory. Homework is also the mechanism for extending the bounds of knowledge and, in that sense, the mechanism is often imperfect by design. In homework, students are challenged up to and often beyond their limits and failure followed by tenacious thought and renewed effort extend those limits.

The completion of homework, then, means attempting every problem assigned and showing all of the work involved in the process.

D. Comment on Showing Work

A prevailing question from students is one questioning why work needs to be shown when correct answers are involved. Students (and parents) need to understand that correct answers are not the ultimate goal in working math problems. [Sounds foolish, doesn't it. Read on.] It is often the case in math that difficult problems are solved using exactly the same concepts and methods as their easier cousins. Working easy problems is training for the more difficult ones. Difficult problems require a high degree of organization, both mental and written, and the tendency to omit steps in easy problems will prevail when more difficult problems are encountered and mistakes will ALWAYS result. Please don't take this lightly.

I've unfortunately seen some promising students unable to progress in math because, at an early age, they found out they had a certain talent to see through math problems and they delighted in showing off by skipping steps. They later find math extremely difficult because the misguided methods were learned so well that they felt overwhelmed and helpless when things got tough. The tendency to skip steps is so pervasive that many students think it's a sign of ignorance and inability when they show the work! How misguided. Show me a student willing to learn the concepts (or general ideas) involved in math and a willingness to think carefully and show the work and I'll show you an engineer.

A good way to encourage students to show work is to tell them that solutions to problems are to be demonstrated in such a way that someone unfamiliar with the problem can easily follow the process. "Show the reader how to work the problem" is a much better approach than simply "find the answer."

E. Homework - Part 2

If homework is to be "graded," then I suggest 2 or 3 points per day to avoid over-weighting the homework in the overall grading scheme. No matter how many points are awarded for the completion of all homework, be prepared to define partial credit, no credit and negative credit (for no homework). I prefer not getting into the fray of defining a homework grading scheme because, in my opinion, it should be avoided except in special cases and those cases are best dealt with by the parents of those students. That said, I will be glad to evaluate and assist in those cases on a one-to-one basis.

The remaining topics involve (1) assigning points to problems on tests, (2) grading tests, and (3) assigning a final grade based upon the accumulation of test scores. Assigning point value and grading tests depend upon an overall grading scheme, so that topic, item (3), appears before the others.

F. Overall Grading Scheme

Of the vast number of grading schemes available, I favor grading on a total- points-scored basis rather than a letter-grade-average basis. That means assigning points to each test without regard to some predetermined total. One test may be worth 74 points and another worth 47. The point total depends on the number of problems on a test and the number of points assigned to each problem.

The total-points-scored scheme is the fairest method of evaluation I have seen because long chapters will naturally have greater weight (because of more problems and more points) than shorter chapters. Therefore, all topics in the text will tend to have relatively equal value.

At first glance, this scheme seems difficult to manage and an interim grade is difficult to calculate. Quite the contrary is true. The trick is to assign letter grades according to a predetermined percent. For example, suppose the grading scale is
A: 92% - 100%
B: 83% - 91%
C: 74% - 82%
Lower grades are not acceptable, implying a need to re-visit topics.

Now suppose the first test is worth 54 points. It's easy to calculate that, for this test, the student must score at least 50 points to make an A. Another way to look at it is this. If the student scores 48 points on the test, then the student's letter grade is calculated as 48/54 = 0.8888...= 89%, which is a B.

Next, suppose on the first three tests the student scores 48/54, 70/74, and 62/66. The calculation of a letter grade is (total points earned) divided by (total possible points) which, in this case, is
(48 + 70 + 62) / (54 + 74 + 66)
180 / 194
0.9278 or 93% which is an A.

This scheme also allows for the addition of points from homework, quizzes, mid-chapter tests, and even extra credit.

G. Assigning Points to Test Problems and Grading Tests

After making out a test, the test should be worked and points assigned to each problem. Every point should be associated with the accomplishment of a particular task within the problem.

For example, suppose this is a test question in a chapter involving formulas:

General Instructions:
Use a formula to solve each of the following problems. Show all work.
Put a box around your final answer.
[The reason for the box is that students sometimes give alternate answers when confused and will later claim the correct one. This may not be necessary for many homeschooled students.]

A homeowner wants to build a large flower garden and put a fence around it. If 150 feet of fencing is available and the garden is to be 30 feet wide, how long must the garden be so that all of the fencing is used?

[A diagram is helpful but is not necessary to solve the problem. Therefore, it is not required and no points are associated with one.]
P = 2L + 2W
150 = 2L + 2(30)
150 = 2L + 60
150 - 60 = 2L
90 = 2L
90/2 = L
L = 45 feet of fencing

I would probably assign 5 points to this problem; 1 for writing the formula, 1 for the numerical part of the answer, 1 for writing the unit of measure, and 2 for the process. Notice that, under certain circumstances, a student could lose just 1 point, just 2 points, and so on. Notice also that the answer is worth only 1 point out of 5 which de-emphasizes the quest for just the answer.

As much as possible, the assignment of point value within a problem should be independent of one another. That idea is certainly true for writing the formula and writing the unit of measure. However, it is less so with the intermediary steps and the answer because surely if a student makes a careless mistake in the process, the answer will almost always be incorrect.

This is where test grading becomes as much art as science. If a student makes a truly careless mistake, like replacing 2(30) with 80 instead of 60, and the other steps are mathematically correct; even though the answer using 80 yields an incorrect value of 35 feet, I would be inclined to take off just one point. I can also make a case for taking off 2 points, but certainly no more.

The key ingredient is consistency. Just realize there will be (hopefully isolated) situations in which you have to split hairs a bit and use your best judgment on the basis of fairness.

When students appear to understand what they are doing and they both show the formula and find the correct answer with the unit of measure, I generally allow some latitude in the number of steps they show. However, I think it's important to be strict about the validity of each equation. That is, in every equation in each step, the value of the expression on one side of the equal sign needs to be the same as the value of the expression on the other.

Sometimes, rather than skip steps, students will "make notes" to themselves within the equation context and, even though they may know what they are doing, the validity of the equation is compromised. For example, suppose you see this in the previous problem:

P = 2L + 2W
150 = 2L + 2(30)
150 = 2L + 60 - 60
150 - 60 = 2L
and so on

Notice in the third step the student is making a note to subtract 60 on the right side. The note was probably made after the writing the correct equation (from above). Subtracting 60 is the correct thing to do next but the equation is technically incorrect in writing -60 only on the right side. Take off a point. The student will therefore realize the importance of the integrity of the equation and will take more care next time.

Score the test questions by writing the number of points subtracted from each problem, if any. Collect those and subtract from the total value of the test. Write the grade at the top of the paper in the form 48/52 A.


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