ERIC Identifier: ED313192
Publication Date: 1989-03-00
Author: Mestre, Jose
Source: ERIC Clearinghouse on Rural
Education and Small Schools Charleston WV.
Hispanic and Anglo Students' Misconceptions in Mathematics.
Math teachers and researchers are beginning to agree on the importance of a
series of new findings. According to new research, many students have
misconceptions about mathematics--sometimes called "naive theories"--that can
turn them into clumsy learners. This digest describes misconceptions in
math--what causes them and why they interfere with learning. Next it considers
common mathematical misconceptions among Anglos and Hispanics. It concludes with
a discussion of techniques to help students overcome their misconceptions in
THE NATURE OF MISCONCEPTIONS
Students do not come to the
classroom as "blank slates" (Resnick, 1983). Instead, they come with theories
constructed from their everyday experiences. They have actively constructed
these theories, an activity crucial to all successful learning. Some of the
theories that students use to make sense of the world are, however, incomplete
half-truths (Mestre, 1987). They are "misconceptions."
Misconceptions are a problem for two reasons. First, they interfere with
learning when students use them to interpret new experiences. Second, students
are emotionally and intellectually attached to their misconceptions, because
they have actively constructed them. Hence, students give up their
misconceptions, which can have such a harmful effect on learning, only with
What do these findings mean? They show teachers that their students almost
always come to class with complex ideas about the subject at hand. Further, they
suggest that repeating a lesson or making it clearer will not help students who
base their reasoning on strongly held misconceptions (Champagne, Klopfer &
Gunstone, 1982; McDermott, 1984; Resnick, 1983). In fact, students who overcome
a misconception after ordinary instruction often return to it only a short time
SOME COMMON MATHEMATICAL MISCONCEPTIONS AMONG ANGLO
There are many misconceptions in elementary mathematics. In fact,
there are so many that researchers have developed a catalog of them (Benander & Clement, 1985). Here, the discussion considers some common misconceptions
that are hard to change.
A very prevalent misconception surfaces in the "students and professors"
Write an equation using the variables S and P to represent the following
statement: "There are six times as many students as professors at a certain
university." Use "S" for the number of students and "P" for the number of
The most common error in this problem (committed by about 35% of college
engineering majors!) is to write "6S = P." (The correct equation, of course, is
"S = 6P.")
This misconception grows from the interplay of two factors. First, students
thoughtlessly translate the words of the problem from left to right. Second, as
they give in to that temptation, they confuse the idea of variables and labels.
Using the left-to-right strategy, students interpret the "S" and the "P" in the
equation as labels (verbal shorthand) for the terms "students" and "professors."
They fail to apply the idea that variables stand for numerical expressions.
A more complex problem also illustrates this type of difficulty:
I went to the store and bought the same number of books as records. Books
cost two dollars each and records cost six dollars each. I spent $40 altogether.
Assuming that the equation 2B + 6R = 40 is correct, what is wrong, if anything,
with the following reasoning?
2B + 6R = 40. Since B = R, we can write,
2B + 6B = 40, and therefore,
8B = 40
This last equation says 8 books cost $40, so one book costs $5.
In this problem students may interpret the letter "B" to mean "books," "the
number of books," "the price of books," "the number of books times the price,"
and other even more imaginative combinations.
"B" is indeed the number of books in the initial equation, but it is also the
number of records in the substitution "B = R." Solving for "B" in the final
equation yields the price of the average item, not the price per book. "B" is a
variable--it stands for an unknown number--not for the word "books."
Teachers will recognize evidence of other misconceptions, as well:
Students often mistake the way in which an original price and a sale price
reflect one another. They often incorrectly calculate the original price from a
sale price by applying the discount to the known sale price, instead of to an
unknown original price.
Students may misconceive the independent nature of chance events. For
example, after getting heads on four consecutive tosses of a coin, they may
claim that tails are more likely than heads on later tosses.
Most college students in remedial math courses want to subtract in the
following problem: "Margaret had 2/3 of a gallon of ice cream. She ate 1/4 of
it. How much ice cream did she eat?" Their misconception shows that they want to
save multiplication for computing increases.
MATHEMATICAL MISCONCEPTIONS AMONG HISPANIC STUDENTS
studies that have investigated mathematical misconceptions among Hispanic
students show that their error patterns are nearly always the result of
differences in language or culture. The discussion that follows presents some of
the unique difficulties experienced by Hispanics, as described in two recent
studies (Mestre, 1982; 1986):
In the "students and professors" problem, Hispanics sometimes write the
answers, "6S = 6P" and "6S + P = T." In the former case, they reason that the
phrase "as many students as professors" implies there is an equal number of
each. In the latter case, students claim that their equation (in which T = total
number of students and professors) combines everyone in the correct proportions.
In both cases, students' misconceptions come from language differences.
Some Hispanics wrote the answer "9 x 28 - 7" to the following problem: "In an
engineering conference, 9 meeting rooms each had 28 participants, and there were
7 participants standing in the halls drinking coffee. How many participants were
at the conference?" The students assumed "participant" referred only to people
in the meeting rooms and that the coffee drinkers came out of the rooms. This
misconception, too, has its roots in language differences.
Some Hispanic students did not grasp this problem: "A carpenter bought an
equal number of nails and screws for $5.70. If each nail costs $.02 and each
screw costs $.03, how many nails and how many screws did he buy?" They believed
it meant the carpenter spent an equal amount of money on nails and screws.
Again, the influence of language is clear.
The number of unique errors among Hispanics resulting from linguistic
difficulties is, however, small. In general, they cause Hispanics to commit the
same types of errors as Anglos, but with a higher frequency. Cocking and Mestre
(1988), in an edited book, deal with this topic among various ethnic and
cultural groups. For example, some chapters discuss the unique mathematical
difficulties experienced by Native Americans and by Oksapmin aborigines of
Papua, New Guinea.
IDENTIFYING AND HELPING STUDENTS OVERCOME
Simply lecturing to students on a particular topic will not
help most students give up their misconceptions. Since students actively
construct knowledge, teachers must actively help them dismantle their
misconceptions. Teachers must also help students reconstruct conceptions capable
of guiding their learning in the future.
Lochead & Mestre (1988) describe an effective inductive technique for
these purposes. The technique induces conflict by drawing out the contradictions
in students' misconceptions. In the process of resolving the conflict--a process
that takes time--students reconstruct the concept.
The following discussion illustrates the three steps of this technique with
the "students and professors" example:
1. PROBE FOR QUALITATIVE UNDERSTANDING. Keep on the look-out for
misconceptions. A simple, well placed question can show if a student's
difficulty comes from linguistic confusion, naive misconceptions, or both. In
the "students and professors" problem, a good question to ask is "Are there more
students or professors in the university?"
2. PROBE FOR QUANTITATIVE UNDERSTANDING. If students understand that there
are more students than professors, the next step is to ask a question (for
example, "Suppose there are 100 professors at the university. How many students
would there be?") In most cases, students will give the answer, "600 students."
3. PROBE FOR CONCEPTUAL UNDERSTANDING. Next, ask students to write an
equation, and look for common error patterns. Now is the time to induce
conflict. For example, with the reversal error, "6S = P," the teacher might ask,
"What would happen if you substituted S=600 in your equation? Would you get
P=100, as before?"
With this inductive approach, the classroom can serve as a forum for some
heated discussions among students who will disagree on an answer. Note that the
teacher does not tell students "the right answer." Instead, the teacher guides
them toward constructing it. In this way, students' most important and most
effective learning has to do with concepts, not just correct numbers. An active
classroom discussion, with the teacher serving as guide, helps students air
their misconceptions and, together, truly overcome them.
Benander, L., & Clement, J. (1985).
Catalogue of Error Patterns Observed in Courses on Basic Mathematics (Internal
Report #115). Amherst, MA: University of Massachusetts, Scientific Reasoning
Research Institute, Hasbrouck Laboratory. (ERIC Document Reproduction Service
No. ED 287 762)
Champagne, A., Klopfer, L., & Gunstone, R. (1982). Cognitive research and
the design of science instruction. Educational Psychologist, 17, 31-53.
Cocking, R., & Mestre, J. (Eds.). (1988). Linguistic and Cultural
Influences on Learning Mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates.
Hardiman, P., & Mestre, J. (in press). Understanding multiplicative
contexts involving fractions. Journal of Educational Psychology.
Lochead, J., & Mestre, J. (1988). From words to algebra: Mending
misconceptions. In A. Coxford & A. Shulte (Eds.), The Ideas of Algebra, K-12
(1988 Yearbook of the National Council of Teachers of Mathematics, pp. 127-135).
Reston, VA: National Council of Teachers of Mathematics.
McDermott, L. (1984). Research on conceptual understanding of physics.
Physics Today, 37, 24-32.
Mestre, J. (1982). The interdependence of language and translational math
skills among bilingual Hispanic engineering students. Journal of Research in
Science Teaching, 19, 399-410.
Mestre, J. (1986). Teaching problem solving strategies to bilingual students:
What do research results tell us? International Journal of Mathematics Education
in Science and Technology, 17, 393-401.
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.
Resnick, L. (1983). Mathematics and science learning: A new conception.
Science, 220, 477-478.