**ERIC Identifier:** ED482727

**Publication Date:** 2003

**Author: **Yetkin, Elif

**Source: **ERIC Clearinghouse for Science Mathematics and Environmental
Education

**Student Difficulties in Learning Elementary Mathematics.
ERIC Digest. **

Learning mathematics with understanding is the vision of school mathematics

recommended by the National Council of Teachers of School Mathematics
(2000). In
order to design and develop learning environments that promote understanding
efficiently, teachers need to be aware of students' difficulties in
learning mathematics.

Drawing from the research in mathematics education, this Digest focuses
on students'
difficulties in learning written symbols, concepts, and procedures
in elementary
mathematics as well as the sources of these difficulties.

**DIFFICULTIES IN LEARNING WRITTEN SYMBOLS**

Standard written symbols play an important role in student learning
of mathematics, but
students may experience difficulties in constructing mathematical meanings
of symbols.
Students derive meaning for the symbols from either connecting with
other forms of
representations (e.g. physical objects, pictures and spoken language)
or establishing
connections within the symbol systems (Hiebert & Carpenter, 1992).
The meaning of
numerical and operational symbols-such as 2, -4, 3/4, 2.4, and +-are
constructed by
connecting with concrete materials, everyday experiences or language.
For example,
the symbol "+" takes meaning if it is connected with the joining idea
in situations like "I
have four marbles. My mother gave me five more marbles. How many marbles
do I
have altogether?" (Hiebert & Lefevre, 1986). Similarly, students
frequently refer to 3/4
as three pieces of a pizza or cake that is cut into four pieces (Mack,
1990).

Although these representations facilitate learning written symbols,
the potential for them
to create understanding of written symbols is limited, since they are
representations
themselves. Students might have difficulty in understanding the meaning
of a written
symbol if the referents do not well represent the mathematical meaning
or if the
connection between the referent and the written symbol is not appropriate
(Hiebert &
Carpenter, 1992). For example, geometric regions are the models most
commonly used to represent fractions. These models represent the part-whole
interpretation of rational numbers. However, the symbol a/b also refers
to a relationship between two quantities in terms of the ratio interpretation
of rational numbers. Similarly, it is used as a way of writing a/b to refer
to an operation. For this reason, teachers need to use other types of representations
such as sets of discrete objects and the number line to promote conceptual
understanding of the symbol a/b.

Most of the difficulty in understanding symbols comes from the fact
that in their standard form, written symbols might take on different meanings
in different settings. For instance, in solving the equation 2x+3=4, x
is regarded as an unknown, which does not vary, whereas it varies depending
on y in the equation 2x+3=y (Janvier, Girardon, Morand, 1993). In order
to understand mathematical symbols, students need to learn multiple meanings
of the symbols depending on the given problem context. Therefore, they
should be provided with a variety of appropriate materials that represent
the written mathematical symbols, and they should also be aware of the
meaning of mathematical symbols in different problem contexts.

Students also build understanding for written symbols by making connections
within the
system. For example, a numeral such as 3254 can express the number
of the units of
any power of ten. In other words, it represents three thousands, two
hundreds, fifty-four units as well as three hundred twenty-five tens; thirty-two
hundreds; and three
thousands. Although these patterns are evident for adults, students
might not easily
construct these relationships by themselves (Hiebert & Carpenter,
1992). Therefore,
teachers should be aware of these difficulties and provide students
with opportunities to recognize the patterns and make connections within
symbol system.

**DIFFICULTIES IN LEARNING ELEMENTARY MATHEMATICAL CONCEPTS AND PROCEDURES**

Students learn new mathematical concepts and procedures by building
on what they
already know. In other words, learning with understanding can be viewed
as making
connections or establishing relationships either within existing knowledge
or between
existing knowledge and new information (Hiebert & Carpenter, 1992).

When students attend school, they have intuitive understanding of many
concepts in
mathematics including numbers, measurements and probability. For example,
kindergarten and first grade students intuitively solve a variety of
problems involving
joining, separating, or comparing quantities by acting out the problems
with collections
of objects (Carpenter & Lehrer, 1999). Extensions of these strategies
can be used as
the basis for developing the concepts of addition, subtraction, multiplication
and division (Carpenter, Fennema, Fuson, Hiebert, Human, Murray, Oliver,
& Wearne, 1999).

Despite their mastery of certain mathematical concepts, however, students
have
difficulty learning elementary mathematics because students are often
discouraged from
using their informal knowledge (Hiebert & Lefevre, 1986; Romberg
& Kaput, 1999).

Mathematics instruction, which does not help students build their formal
knowledge on their informal knowledge, may cause students to develop two separate
systems of mathematical knowledge.

It is interesting to note that students who obtain incorrect answers
for their written
calculation are often able to find the correct answer by using concrete
materials. However, when they are confronted with their written work, about half
of these students kept their incorrect answer for written work. This discrepancy
between the results obtained from working in two different settings reveals
that students often cannot make connections between formal and informal
mathematics (Lesh, Landau, & Hamilton, 1983). In another study, fourth
graders who had connected decimal fraction numerals with physical representations
of decimal quantities were more successful in dealing with problems that
they had not seen before-such as ordering decimals by size and changing
between decimal and common fraction forms-than students who had not made the same connections (Wearne & Hiebert, 1988). For these reasons,
teachers should provide context to help students bring about their intuitive
mathematical concepts and procedures, encourage them to argue whether they are reasonable,
and guide them to make connections between their intuitive and formal mathematical concepts and procedures (Lampert, 1986).

Student errors are often systematic and rule-based rather than random
(Ben-Zeev,
1996). In addition to student inventiveness, these errors may be caused
by instruction
that focuses on rote memorization. Students abstract or generalize
procedures from
following the steps in worked-out examples, but when their knowledge
is rote or
insufficient, they might overgeneralize or overspecialize the rules
and procedures
(Ben-Zeev, 1996; VanLehn, 1986). For instance, students might overgeneralize
the rule for subtracting smaller from larger on single-digit subtractions
to multidigit subtraction problems if they are only taught to subtract
the smaller from the larger number. Similarly, if students are exposed
only to borrowing two digit subtractions, they may overspecialize borrowing
from the units-digit to multi-digit subtractions (VanLehn, 1986).

One way to reduce such difficulties is to help students make connections
between
conceptual and procedural knowledge. The construction of conceptual
knowledge
requires identifying the characteristics of concepts, recognizing the
similarities and
differences among concepts according to these characteristics, and
constructing the
relations among them. On the other hand, procedural knowledge requires
constructing
skills, strategies or algorithms that are means to an end (Byrnes &
Wasik, 1991). For
example, students who do not align decimal points while adding or subtracting
decimal
fractions probably follow the algorithm without making connections
between position
values of decimals and lining up the decimal points (Hiebert &
Lefevre, 1986). More advanced connections, such as adding up values that are alike, require
generalizing
and reflecting on pieces of information such as lining up decimal points
in order to add
decimal fractions or looking for common denominators while adding common
fractions.

Although such connections might be obvious for adults, constructing
them might be
difficult for the students. Teachers need to design instruction that
helps students
construct these big ideas. With regard to classroom instruction, student
difficulties can also be attributed to using inappropriate representations.
For example, students having difficulty in adding fractions may extrapolate
erroneous algorithms from instruction on the representation of fractions.
Students who are often presented fractions by using pie graphs perform
"1/2 + 1/3 = 2/5" and justify the solution as "adding one piece of a two
piece pie and one piece of a three piece pie will result in two pieces
out of five pieces altogether" (Silver, 1986; Ben-Zeev, 1996). As discussed
earlier, using appropriate representations will help students construct
different characteristics of concepts.

**CONCLUSION**

Developing understanding in mathematics is an important but difficult
goal. Being aware
of student difficulties and the sources of the difficulties, and designing
instruction to
diminish them, are important steps in achieving this goal. Student
difficulties in learning
written symbols, concepts and procedures can be reduced by creating
learning
environments that help students build connections between their formal
and informal
mathematical knowledge; using appropriate representations depending
on the given
problem context; and helping them connect procedural and conceptual
knowledge.

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