Introduction
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. Testtaking 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 oddnumbered
problems, you may wish to limit your choices to the evennumbered 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 evennumbered
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 subsections,
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 midchapter quizzes and those may be used as a source of
test questions when deciding to test at midchapter 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, midterm 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
overweighting 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 onetoone 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
pointsscored basis rather than a lettergradeaverage 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 totalpointsscored 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 revisit 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,
midchapter 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.]
Problem:
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?
Solution:
[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 deemphasizes 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.
