Suppose that an algebra problem asks a student to solve an
equation for an unknown. The goal of the problem is to test whether or not the
student knows how to apply a particular algebraic technique.
However, if the question is multiple choice, the student
doesn’t actually need to know how to solve the equation in order to determine
the correct answer. The student could simply plug each answer into the equation
to see which one works.
Example: 3x + 20 = 12x + 2.
(A) x = 1 (B) x = 2 (C) x = 3 (D) x = 4 (E) x = 5
A student could simply try each answer. Plugging in x = 1,
it’s easy to see that 23 doesn’t equal 14. Trying x = 2, 26 = 26. The student
already has the answer, but hasn’t done any algebra.
If the correct answer is (E), or if the student makes a
mistake in the calculation, it will take longer, but in principle, a student
can get the correct answer without doing any algebra.
If there are fractions in the answers, that makes the
calculation a little more cumbersome. Many students prefer to avoid fractions,
so this may deter students from avoiding the algebra.
Suppose that the choices had been:
(A) x = 1/2 (B) x = 18/15 (C) x = 22/15 (D) x = 2 (E) x =
22/9
Some problems can be solved faster by actually doing the
algebra. But more time-consuming problems, like using the quadratic or solving a
system of equations, might be solved faster by just plugging the answers into
the original equations. When algebra yields the fastest solution, this provides
an incentive to solve the problem as intended.
If a student relies on plug and chug for every problem, the
student risks not finishing the test. But students probably won’t try to solve every problem this way. Just the hard
ones.
Fortunately, there are many problems that tests can ask that
can’t be solved this way. For example, many problems ask students to simplify
an expression, and there are also conceptual, strategic, and logical questions.
This helps to ensure that students must grasp some algebra concepts in order to
pass the class.
The main goal of an algebra course is for students to become
fluent in solving for unknowns in a variety of types of equations. Multiple
choice is convenient, especially in large classes. It’s worth considering
whether or not students might find ways to succeed in the course without
actually mastering the techniques.
One way is for the instructor to become acquainted with the
students, perhaps by checking written solutions of homework or quizzes (or
adding a few written problems to exams, where feasible), helping students solve
problems on the board, or interacting with students during office hours. If
students who ordinarily struggle solving equations are better able to figure
out the right answers on multiple choice problems, this might be a signal (of
course, this could also be the result of studying, tutoring, and improvement).
If instead experience with their written solutions corresponds well with their
ability to solve similar problems on multiple choice exams, then there may not
be any reason to worry that the students who most need to improve their fluency
might be finding an easy way out (which may very well be the case for many of
the students – it might be the clever problem-solvers who are most likely to
think of this).
Chris McMullen, author of the Improve Your Math Fluency series of workbooks
