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Common Misconceptions in Basic Mathematics
by
G. Donald Allen
Department of Mathematics
Texas A&M University
College Station, Tx 77843
In this note we look at typical mathematical misconceptions. Some are rooted in arithmetic and
some in algebra. These misconceptions have many origins, learning incorrectly, informal
learning, and mis-remembering, to name a few. Many are based in arithmetic, but some seem
to enjoy their source in algebra. As well, some arise in the misconceptions about the order of
operations.
What is a misconception?
It may be best to begin with a definition. From the Encarta online dictionary, a
misconception is “a mistaken idea or view resulting from a misunderstanding of something."
Paraphrasing from the educational literature [Pines, 1985] , we obtain,
Certain conceptual relations that are acquired may be inappropriate within a certain context. We
terms such relations as "misconceptions." A misconception does not exist independently, but is
contingent upon a certain existing conceptual framework. Misconceptions can change or disappear with
the framework changes.
Changing conceptual framework is one of the keys goals in repairing mathematics and science
misconceptions. That is to say, it is not usually successful to merely inform (e.g. lecture) the
student on a misconception. The misconception must be changed internally partly through the
student’s belief systems and partly through they’re own cognition.
In another misconceptions framework, we may say many students do not come to the
classroom as "blank slates" (Resnick, 1983). Rather, they come with informal theories
constructed from everyday experiences. These theories have been actively constructed. They
provide an everyday functionality to make sense of the world are often incomplete half-truths
(Mestre, 1987). They are misconceptions.
In this note, we consider student misconceptions in mathematics, particularly those that
impact algebra and algebraic thinking. Yet, misconceptions are but one facet of faulty,
inaccurate, or incorrect thinking. These are all intertwined causing students unlimited trouble
in grasping with mathematics from the most elementary concepts through calculus. In turn,
they cause teachers immense frustration about why their teaching isn’t "getting through." We
consider here the basics of misconceptions including issues of cancellation, fractions,
arithmetic, functions, and powers.
Misconception Examples
Cancellation. Some students do not cancel quantities between the numerator and denominator
correctly, often creating ad hoc rules along the way.
 Consider the evaluation of the fraction 844  21. What some students do is simply cancel
off the number 4 coming up with
84  8 (Incorrect)
4
It is not clear the origins of this misconception, except possibly with algebra, where (with
b ≠ 0)
ab  a
b
is exactly correct. Here we have a misconception that arises when students apply a rule
for variables to a problem involving numbers.
 Let’s look at the perimeter formula for a rectangle, P  2W  2L. Find W. We have
P − 2W  2L
P − 2W  L
2
P − W  L (Incorrect)
This is another example of an erroneous cancellation.
Fractions. There are several classics concerning fractions.
 Addition of fractions. Adding the numerators and the denominators is clearly incorrect
but many students do this as a matter of recourse.
1  3  4 (Incorrect)
7
2
9
This is of course wrong. With all the practice students have at adding fractions, it is
amazing students continually give such answers. However, when we look at
a  c  ad  bc
b
d
bd
we see the complexity of this answer is difficult to remember. Note the use of the word
"remember." It is important for students to fully understand that adding fractions required
finding equivalent fractions having the same denominator. Relying on memory is
tantamount to relying on faulty memorization.
 What should we do with 0? Many students will express
5  0 (incorrect)
0
However, most students are confident that 05  0. So, is the incorrect version a confusion
or a misconception? In either case, it leads to wrong answers.
 If we must compute 32  15
many students will write this as
2
5
3 
2
5
3
2
5
 3
10
(incorrect)
This illustrates the misconception of the "invert and multiply" rule. The relative sizes of
the fraction bars may play a role in this misconception. Similarly, some students will
express
3
2
1
7
 3  1  3
2 7
14
(incorrect)
illustrating another misconception with the "invert and multiply" rule.
Arithmetic. It is important for students to know exactly the rules of operations.
 What is the value of −3 2 ? While we know the answer is −9, many student will write
− 3 2  9 (incorrect)
forgetting the order of operations requires that powers precede multiplication, in this case
multiplication by (−1. I don’t know of any way to teach this except by the repetition of
the instruction and the drill of rules.
 What is 3 2  4 2 ? It is clear the answer is 5, but some students will write
3 2  4 2  3  4  7 (incorrect)
There is one way to avoid your students from doing this, and that is by asking them to
roughly compute the value, in this case 3 2  4 2  9  16  25  5. To uncover
this type of misconception, it is important for students to discover themselves their idea is
wrong. Many folks in the professions of math and engineering, and other directions make
conclusions erroneously. When they are careful they check for the plausibility of their
answer.
 Find 3 − 4 − 5  4. This is a simple calculation. But some students will write this as
3 − 4 − 5  3 − 4 − 5 (incorrect)
 −6
forgetting to distribute the minus sign fully. Of course some students may compute
4 − 5  −1, giving the result 3 − 4 − 5  3 − −1  4. When rules are not thoroughly
understood, or at least memorized, other possible answers are opened for review. You’ll
note that one logic works while the other does not. Also, from 3 − 4 − 5, we could
compute 3 − 4 − 5  − 6, yet another way to achieve the incorrect answer. A rule of
thumb is that whenever an expression involving only pluses and minuses is involved,
unless appropriate delimiting parentheses are used, all results are suspect.
Algebraic Expressions. Many algebraic misconceptions abound.
 Solve 4 − x − 2  8. We know the solution is x  −2. However what some students do
is write
4 − x − 2  8
4 − x − 2  8 (incorrect)
2−x  8
x  −6
The failure to use the distributive law for multiplication is evident. Had we posed the
problem 4 − 2x − 2  8, many student would distribute correctly giving the answer of
x  0. When the multiplier is not indicated explicitly, therein lies a possible
misconception about what to do.
 Here is another common misconception, also involved with rules on the order of
operations. Solve 5  2x  4  12, which we know has the solution x  − 12 . Some
students will add the 5 and the 2 to get
7x  4  12
. this is an interesting situation. When a student sees this
which has the solution x  − 16
7
16
answer − 7 , they may be a bit put off or at least suspicious. If this is a multiple choice
item on a test, and the student sees the incorrect answer, he/she is likely as not to select it.
which has no reduction
 Combining algebra and fractions students asked to evaluate 4x−3
4
will often cancel the four’s obtaining
4x − 3  x − 3
4
A more general version of the same thing is
ab  c to obtain a  c
b
We see this type of misconception often in algebra.
 For the expression 2x1
, which has no reduction, some students will simply cancel the
3x5
x-terms and reporting
2x  1  2  1  3
3x  5
35
8
 Relatively rare is this type of misconception involving the solutions of a quadratic. Solve
x 2  x  −3, which has the solutions 12 i 11 − 12 , and − 12 i 11 − 12 . However, it is
possible to see the following
x 2  x  −3
xx  1  −3 yielding
x  −3 or x  −4 (incorrect)
This ad hoc rule for solving a quadratic is simply incorrect and illustrates a profound
misunderstanding of the quadratic rule.
Functions. Working with functions and algebraic expressions often reveals many difficulties
for students. Indeed, if you need to know just how much your students understand about these
constructs consider the following problems.
 When asked to compute x − 5 2  x − 5 2 , some students will show
x − 5 2  x 2 − 5 2 (incorrect)
This misconception is rather rare as students are often taught "by the book" how to square
such an expression.
 Calculus students often misstep on this type of problem. Let fx  x 2  3. Find fx  h.
First the computation is exactly fx  h  h 2  2hx  x 2  3. What many students do is
the write following
fx  h  x 2  3  h (incorrect)
Indeed, they persist in this regardless of many times they are instructed it is incorrect.
 The next example is similar, but with a different error. For the function given above find
fx  h − fx  h  x 2 − x 2  h 2  2xh. This correct result is not often what we see.
Often, the answer given is
fx  h − fx  x  h 2  3 − x 2  3
 x 2  2h  h 2  3 − x 2  3 (incorrect)
 h 2  2h  6
This issue was considered above with numbers.
fxh−fx
 Finally, as a precursor to calculus, students are asked to compute
. We know the
h
answer is
fx  h − fx
 2x  h
h
But if the student has made either of the two incorrect answers above, he/she will surely
arrive at the incorrect answer. These examples show the necessity of students executing a
sequence of operations correctly.
 Domain problems are often a source of misconceptions. For example, find the domain of
1
fx 
3x  1
1
We know that 3x  1  0, or x  − 3 . Some students will recognize only that the quantity
3x  1 ≠ 0, and thus report the domain to be x ≠ − 13 , forgetting completely the radical in
the expression.
Powers. Students make many mistakes or have many misconceptions about powers. Again,
the culprit seems to be with full and complete understanding of the rules, and the ability to
carry them out correctly.
3
 Compute xx−2  x 5 . Some students will write
x 3  x 3−2  x (incorrect)
x −2
indicating a faulty knowledge or misconception about what to do with negative powers in
the denominator. It seems that some may know that when bringing a power from the
denominator to the numerator there must involve a change of sign, but then forget the
power in the denominator is also negative.
 In another example the student is asked to compute 2x 2  3  8x 6 will generate the answer
2x 2  3  2x 6 (incorrect)
with the misconception that the cube power must apply to both multiplicands within the
parentheses. On the other hand the student may write
2x 2  3  2x 5
or even 2x 2  3  8x 5 , adding the powers rather than multiplying them. This
misconception is two more common answers given by students working with powers of
numbers.
 When it comes to radicals, many students make common mistakes owing partially to
misconceptions and also incorrect instruction. Here is the most frequent error.
−6 2  −6 (incorrect) or
−6 2  6 (incorrect)
is invoked, the correct mathematical definition is to take the positive
In fact, once the
radical, if it exists. So −6 2  36  6.
 The most frequently occuring misconception is
a2  b2  a  b
 Another common misconception is to incorrectly factor and then extract radicals.For
instance, some students will conclude that
4x 2 − 4x  4 x 2 − x (incorrect) or
4x 2 − 4x  2x − x x 2 − x (incorrect) or
The correct answer for this one is 4x 2 − 4x 
4x 2 − x  2 x 2 − x , the term
x 2 − x cannot be further reduced.
Conclusions
Most of the examples above and the incorrect answers offered by students indicate a lack of
precise understanding of the rules, though some of the errors are more serious. It is also clear
that those based on arithmetic can cascade to higher level mathematics courses. Moreover, it is
not clear how inquiry based instruction can help with these. Yet, it is possible. It is certain
that once students have misconceptions about mechanical mathematical operations, they are
difficult to remove, if not impossible. There seems no systematic effort at teaching students or
pre-service teachers away from misconceptions. To my knowledge, there are no courses on
this topic.
References
1. Gourgey, Annette F, The Relationship of Misconceptions about Math and Mathematical
Self-Concept to Math Anxiety and Statistics Performance, Not published. General
information on misconceptions.
2. Champagne, A. B., Gunstone, R. E, & Klopfer, L. E. (1983). Naive knowledge and
science learning. Research in Science and Technological Education, 1(2), 173-183.
3. Clement, John. (1982) Algebra Word Problem Solutions: Thought Processes underlying a
common misconception, Journal for Research in Mathematics Education, 13, 16-30.
4. Confrey, Jere. (1982). A Review of the Research on Student Conceptions in Mathematics,
Science, and Programming, Review of Research in Education, Vol. 16, pp. 3-56.
5. Ginsberg, H. (1977). Childrens’s arithmetic: How they learn it and how you teach it.
Austin, TX: Pro-Ed.
6. McDermott, L. (1984). Research on conceptual understanding of physics. Physics Today,
37, 24-32.
7. Mestre, J. (1987, Summer). Why should mathematics and science teachers be interested in
cognitive research findings? Academic Connections, pp. 3-5, 8-11. New York: The
College Board.
8. Osborne, R. J., & Wittrock, M. C. (1983). Learning science: A generative process. Science
Education, 67(4), 498-508.
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A. L. Pines (Eds.) Cognitive structure and conceptual change (pp.101-116). New York,
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