Creative Education
2010. Vol.1, No.3, 138-148
Copyright © 2010 SciRes. DOI:10.4236/ce.2010.13022
Language and Mathematics: Bridging between Natural Language
and Mathematical Language in Solving Problems in Mathematics
Bat-Sheva Ilany1, Bruria Margolin2
1Beit-Berl College, Israel;
2Levinsky College of Education, Israel.
Email: bat77@013net, b
Received June 21st, 2010; revised August 25th, 2010; accepted September 18th, 2010.
In the solution of mathematical word problems, problems that are accompanied by text, there is a need to bridge
between mathematical language that requires an awareness of the mathematical components, and natural lan-
guage that requires a literacy approach to the whole text. In this paper we present examples of mathematical
word problems whose solutions depend on a transition between a linguistic situation on one side and abstract
mathematical structure on the other. These examples demonstrate the need of treating word problems in a litera-
cy approach. For this purpose, a model for teaching and learning is suggested. The model, which was tested
successfully, presents an interactive multi-level process that enables deciphering of the mathematical text by
means of decoding symbols and graphs. This leads to understanding of the revealed content and the linguistic
situation, transfer to a mathematical model, and correspondence between the linguistic situation and the appro-
priate mathematical model. This model was tested as a case study. The participants were 3 students: a student in
the sixth grade, a student in the ninth grade and a college student. All the students demonstrated an impressive
improvement in their mathematical comprehension using this model.
Keywords: Mathematical Language, Word Problems
Natural Language and Mathematical Language
In the solution of word problems, that is to say in the solution
of mathematical problems that are accompanied by text, the
student is faced with two languages mixed together (Kane,
1970): natural language and mathematical language. One of the
limitations of natural language is the fact that it works in a dia-
chronic manner; the meanings it represents are interpreted ac-
cording to a time frame. The perception of the world, and in
fact the meaning of the world, depends on field synchronicity,
the reciprocal relations with the environment. Furthermore, the
field as a whole gives meaning to its parts, and each part of the
field contributes meaning to the whole. In other words, to read
a word problem in mathematics and to give it meaning, it is
necessary to perceive the problem as a textual unit and not as a
collection of data.
A textual unit is a linguistic unit larger than a sentence. Its
semantic structure “consists of those elements and relations that
are directly derived from the text itself [...] without adding an-
ything that is not explicitly specified in the text” (Kintsch, 1998,
p. 103). The linguistic unit is coherent from a content and lin-
guistic point of view (Brown & Yule, 1983; Halliday & Hassan,
1976; Van Dijk, 1980; Widdowson, 1979), and is used for
communication (Sarel, 1991, Margolin, 2002). Identification of
the constituent parts of a text depends on meta-a w areness of the
function of the form, the function of the word, and the function
of the sentence in a text, especially awareness of symbols and
syntactic awareness (Herriman, 1991 in MacGregor, & Price,
1999). Indeed, exercises like addition and subtraction or mul-
tiplication and division are important for the understanding of
the mathematical language, but the perception of the textual
structure is a process by which you can identify textual com-
ponents and carry out different logical operations.
Mathematical language is a language of symbols, concepts,
definitions, and theorems. Mathematical language needs to be
learned and does not develop naturally like a child’s natural
language. In mathematical language the child learns to recog-
nize, for example, numbers as objects, one to one of their simi-
lar and different properties. The child perceives the numbers as
signs by means of which it is possible to calculate calculations
and to do various manipulations.
Syntax, generally, deals with configuration rules according to
which sentences and words are constructed. The syntax of ma-
thematical language includes lists of symbols, configuration
rules for constructing language patterns, axioms, a deductive
system, and theorems. Mathematical terms and symbols must
be defined unambiguously. Likewise, every assertion in mathe-
matical language is also unambiguousevery mathematical
pattern has one deep structure that is determined by operational
r ule s.
We will not expand on definitions and theorems in mathe-
matics, but each definition of a mathematical concept is the
result of a complex process. Each definition contains additional
concepts that also need to be defined. All mathematical theo-
rems in all branches of mathematics are characterized by the
fact that they are derived logically, deductively, and consis-
tently from a system of elementary theorems a xioms.
Our central argument is that there is a bridge between ma-
thematical language that necessitates seeing the mathe ma ti c-
cal components, and natural language that demands textual
literacy for the text as a whole. In other words, there is a bridge
between the mathematical components and the literal compo-
nents. When knowledge gaps in the mathematical language are
large, natural language must supply what is missing, and be
clear and explicit. However, when knowledge gaps in the ma-
thematical language are small, natural language does not need
to supply what is missing.
Differences between Natural Language and
Mathematical Language
The fundamental differences between natural language and
mathematical language derive first and foremost from the fact
that mathematical language is more precise and less flexible
than the structure of natural language. In natural language there
are differences between surface structure and deep structure of
the utterance, there are ambiguous statements that derive from
ambiguous words, and there is a wealth of language. This
wealth of language derives from the diversity of nouns, and
from the diversity of words expressing relation - all the verbs
and adjectives. On the other hand, in mathematical language for
each surface structure there is one deep structure, all statements
are unambiguous, and there is a paucity of language that ex-
presses itself in the fact that there is only one type of noun
numbers, functions, etc., and that there are two relational signs
equality and inequality (Bloedy-Vinner, 1998).
Different concepts are interpreted in the different languages
natural language and mathematical language in different ways.
The structure of mathematical language is more precise and less
flexible than the structure of natural language, thus great tension
is created in the use of natural language in mathematical problems.
Let us take for example the concept diagonal. Diagonal in
mathematical language is a straight-line segment that joins two
non-adjacent vertices in a polygon. The diagonal can also pass
through points outside the polygon, and can be at different
angles. On the other hand, in natural language no mathematical
rules apply to the diagonal and as evidence we can give as an
example the following advert.
According to the advert, it is forbidden to cross the road on a
diagonal that is at an angle different from 90 degrees between
the crossing and the sidewalk. The diagonal in the advert is not
a mathematical diagonal, since it does not join two vertices of a
Like the diagonal, the straight line is also different in mathe-
matical language than in natural language. Whereas in natural
language a straight line is a segment with a beginning and an
end, in mathematical language it is a fundamental undefined
concept, with no beginning and no end.
Translating from Natural Language to
Math ematical Language in Mathematical
Word Proble ms
A mathematical problem is a situation in which an individual
or a group of people are asked to perform a task for which there
is no immediately available algorithm that completely defines a
solution method. Solving mathematical problems requires the
imp lementation of a sequence of operations by means of which
some final target is attained (Lester, 1978).
A word problem in mathematics is an independent unit of
text that comprises a question sentence and a speech event
(Nesher, 1988; Nesher & Katriel, 1977). Sometimes the unit of
text describes an event from daily life. The aim of the descrip-
tion is to give expression to the logical structure that dictates a
particular arithmetic operation. The difficulty with the solution
of mathematical word problems is the need to translate the event
described in natural language to arithmetic operations ex-
pressed in mathematical language. The transition from natural
language includes syntactic, semantic, and pragmatic under-
standing of the discourse.
Word problems in mathematics are divided into two types
according to the topics they relate to. One type is mathematical
word problems that deal with mathematical relationships
between objective sizes, like: What is the number that is twice
as big as the sum of 25 and 17? The other type is mathematical
word problems that deal with real life situations, like: It takes 3
workers 5 hours to plough a field. How many hours would it
take 2 workers to plough the same field?
In this paper we will relate to every mathematical problem
accompanied by text as a word problem in mathematics. There
is a tendency in the professional literature to relate to a word
problem as a textual unit that describes an everyday event, and
the majority of papers and researches relate to problems not
accompanied by an authentic background story as mathematical
problems and not as word problems. Thus, this problem: “Find
the equation of the straight line that is parallel to the straight
line 3x 7y = 4 and passes through the point (0, 10)” is not
considered to be a word problem, rather a mathematical problem,
since it is not accompanied by an authentic background story.
An example for this could be Polya’s treatment in his book
How to Solve It (1945) of mathematical problems accompanied
by text as problems, although at the same time he suggested
that for the problem to be understood it is first of all necessary
for the verbal version to be understood. In our opinion, those
problems must be defined as word problems since such
problems include text that the solver needs to understand.
We attempted to implement the teaching and learning model
proposed in this paper with two kinds of mathematical problems:
one, accompanied by an authentic background story that is
considered to be a word problem in the professional literature;
and another, not accompanied by an authentic background story,
that is not considered to be a word problem in the professional
literature. We argue that a mathematical problem with the
follow ing properties is a word problem even when it is not
accompanied by an authentic background story: it constitutes a
coherent textual unit in terms of content and language; it has
clear boundaries; it is used for communication; it contains two
different languages mixed together: natural language and
mathematical language (see the case of Shiri later).
Following are examples of mathematical word problems, in
which the solution of the problem depends on converting a
linguistic situation into a mathematical model. These examples
differ from each other in many varied aspects.
The Soldiers and the Bus Problem
Translation from natural language to mathematical language
in mathematical word problems is problematic also because of
the difference between solving an authentic realistic problem
and solving a word problem in mathematics. An example of
this is “The soldiers and the bus problem”:
An army bus can transport 36 soldiers. 1128 soldiers need to
be bused to training camp. How many buses are needed?
The appropriate calculation is: 1128:36 = 31(12). The answer
to the problem is 32 buses. 12 soldiers will go in the 32nd bus
(or less than 36 soldiers will travel in some of the buses bus).
In a research that was conducted among 13-year-old students
only 23% answered this correctly, 19% answered 31 buses, and
29% gave as the answer 31 buses remainder 12 (Silver, Shapiro,
& Deutsch, 1993). Contrary to the previous problem, where
there was an attempt to connect the answer to the real situation,
in this problem there is no such attempt. Apparently this is be-
cause of the fact that the arithmetical solution is easily arrived
at, and thus there is no need to connect the problem to the real
The Sheep and the Dogs Problem
Students make the transition from natural language to mathe-
matical language automatically even in problems that do not
have a mathematical solution, because the norm in school is
that if a problem is given, then it must have a solution. For
Five dogs guard 125 sheep. How old is the shepherd? (Baruk,
Students try to find a calculation, and they try to solve the
problem through trial and error. They try to check different
possible solutions by using the given numbers and ignoring the
situation. They try solutions according to arithmetic operations,
from the easy to the difficult, as follows:
1. By adding: 5 + 125 = 130
2. When the number seems to them to be too big to be a
person’s age, the students try to solve the problem by
subtracting: 125 5 = 120
3. When this number is also seems to be too big, they try to
solve the problem by dividing: 125: 5 = 25. This solution seems
to them to be reasonable.
The Students and the Professor Problem
An example of a mathematical problem and of the difficulty
of translating from natural language is The Students and the
Professor Problem (Kaput & Clement, 1979). This problem,
which exposed the reversal in translation mistake, has been
investigated in various researches (e.g., Clement, 1982; Ros-
nick, 1981), and has been explained in different ways (Bloedy-
Vinner, 1998).
Write an algebraic expression using the variables S and P
that will represent the following assertion: “In this university
the number of students is 6 times greater than the number of
professors”. Use S for the number of students, and P for the
number of professors.
When this problem was given to 150 first year engineering
students and 47 social science students, 37% of the engineering
students and 57% of the social science students got it wrong.
Two thirds of the wrong answers contained reversed equations
of the type 6S = P instead of 6P = S. The mistake derived from
the fact that in natural language the order of the words deter-
mines the meaning. The quantifier 6 appears beside the noun
(students) and equality in the mathematical equation is not con-
sidered. According to natural language, the students were
represented by S, the arithmetic operation was represented by
multiplication, and finally the professors were represented by P,
and thus 6S = P.
The problem of the students and the professor was given to
eight fourth-year students specializing in mathematics for ele-
mentary school at a teachers’ training college in Israel. Apart
from one student who solved it correctly, all the other students
erred, writing 6S = P instead of 6P = S. Here also, as in the
a bove-mentioned cases, the mistake arose from the fact that in
natural language the order of the words determines the meaning.
The quantifier 6 appears beside the noun (students) and equality
in the mathematical equation is not considered.
The Reduced Number Problem
Another example of a mathematical problem for which there
is a need for transition from natural language to mathematical
language is “The number that was reduced”:
Write an equation, using the variable X, which represents the
following assertion: “In this college a quarter of the number of
students, which was reduced by 5, are mathematics students”.
When this problem was given to students of mathematics
education in a teachers’ training college, the following expr es-
sions were given:
The two above expressions are correct. The difference be-
tween them derives from the fact that the surface structure ex-
pressed in the assertion represents two deep structures. For each
deep structure there is a different mathematical expression. In
the first expression, the quarter was of the number of students
in the college reduced by 5. Whereas in the second expression,
the quarter was of all the students in the college, and only then
was the number reduced by 5. Both answers are correct.
The Combinations Problem
Another example of a word problem for which misunder-
standing the literal meaning caused a misunderstanding of the
mathematical problem is Com binations (Woolf, 2005). The
problem was presented during a fourth grade mathematics les-
s on:
In a parking lot there are 66 wheels, some of them belonging
to cars, and some of them belonging to tricycles. What combi-
nations of cars and tricycles can you find that have 66 wheels
altogether? How could you know whether you have found all
possible combinations?
The combinations problem documented in Woolf’ s paper
describes students’ inability to cope with the problem due to the
fact that they did not understand the word combinations. Solu-
tion of the mathematical problem described above focused on
understanding the word combinations and understanding the
logical structure of the whole text. To understand the problem,
students are required more than command of the language.
They need to learn to construct a meaningful body of knowl-
edge from the information in the question, including data and a
1This problem was given in different countries (e.g., Norway, Japan,
and Ireland) and in different populations, and the results were similar
(Greer, 1997).
solution scheme. In other words: they need to make a connec-
tion between natural language and mathematical language.
Towards Bridging between Natural Language
and Mathematical Language
As we saw above, the knowledge gaps between mathematical
language and natural language emerge especially in the solution
of word problems. Problem solving is an all-encompassing
concept that includes many cognitive processes like, among
others, literal and syntactic processing, changes of represe nta-
tion, and algorithmic processing. Representation is an important
area for problem solving and little is known about the relation-
ship between inner comprehension and its external representa-
To bridge between natural language and mathematical language
it is necessary to educate towards mathematical-linguistic
literacy2, at the level of the addressor (speaker or writer) and at
the level of the addressee (listener or reader). From the point of
view of the addressor, he/she must ensure that the references in
the text are related to suitable referents, that the expressions are
not ambiguous, and that all the problematic terms are made
clear. In other words, the addressor must display consideration
for the recipient by supplying easily accessible and acceptable
information. The addressor must take into account that the
meaning of the text is a product of the interactions between the
addressor’s schema and deductions, and the addressee’s schema
and conclusions. Thus the addressor must first determine precise
assumptions about the addressee’s knowledge and deductive
abi l ity, to produce text as explicit as possible by means of
linguistic implementation of his/her ideas. The addressor must
predict detractors and obstacles that are liable to disturb
understanding, and take steps to prevent them (Folman, 2000).
As for the addressee, to extract the full meaning from the text
he must fill in the missing information that is not found in the
text. In other words, he/she must identify the information in the
text in three circles of connection. The adjacent and literary
connection (co-text) is the connection formed inside adjacent
linguistic units. The circumstantial pragmatic connection
(context) includes different components like the identity of the
addressor and the addressee, the time and place of the discourse,
the addressor’s intentions, and the communication medium (Nir,
1989). The connection with the universe of discourse that is in
fact the connection formed between the text and the world
(Sarel, 1991), is based on our previous knowledge of the
particular subject area.
The knowledge gaps in problem solving are between the
textual unit and the hidden mathematical structure. The linguistic
units in the text not only function as signs that have their
signified object or idea in the world external to linguistics but
also are connected to other fundamentals in the text, so that
their meanings arise from the way the linguistic components are
organized in the text. Moreover, not all the information is given
explicitly in the text. There is information that can be derived
by mathematical means on the basis of the explicit information.
To bridge the gaps between natural language and mathe-
matical language it is necessary to develop an awareness of
these gaps by means of meaningful interactions between the
student and his/her environment in the context of authentic
activities. These authentic activities are activities that represent
situations that describe a reality that is connected to the world
of students and teachers in school and in the community (Ben-
Chaim, Keret, & Ilany, 2006). A language develops by means
of meaningful interactions between the individual and his/her
environment in the context of authentic activities (Gee, 1996).
The context of authentic activities enables education towards
linguistic-mathematical literacy, because the interpr e ttation of
the discourse is based to a great extent on analogy to our past
experience, meaning that it is based on our socio-cultural
knowledge (Brown & Yule, 1983).
According to Greer (1997), the concepts addition, subtraction,
multiplication, and division forcefully create models for
situations, and students have to differentiate between the model
and the situation and to assess whether the model is appropriate
to the situation. The process of creating meaning for the context
is multi-layered and is carried out at every layer of the text: at
the syntactical, the semantic, and the pragmatic layers, at
different levels of focusing.
The interaction between the reader’s schemas and the schemas
of the text necessitates communicative-cognitive effort. The
communicative effort is expressed by the identification of the
situation described, and the cognitive effort is expressed by the
composition of the problem from new while combining within
it the mathematical model.
Bridging between natural language and mathematical language
necessitates connecting the two faces of the word problem: the
linguistic situation on one side and the abstract structures on the
other (Greer, 1997). According to the professional literature,
the bridging can be carried out in two different ways: by
translation of the linguistic situation into abstract structures
(Polya cited in Reusser & Stebler, 1997), and by organization
of the unit of mathematical content (Freudental, 1991). In this
paper we suggest making an interaction between the two
methods in a processive approach by a model for the Instruction
and Learning of the Solution of Mathematical Word Problems.
Many researches that deal with varied subjects in the learn-
ing of mathematics like, for example, real mathematics (De
Lange, 1987), dilemmas in mathematics instruction based on
different representations of concepts (Ball, 1993), solution of
verbal questions (Nathan, Kintch, & Young, 1992), and learn-
ing concepts like the concept of function (Kaput, 1993), main-
tain that development of modeling skills is one of the important
aims of a mathematics curriculum, and serves as a central pe-
dagogical tool.
Instruction Model for the Solution of
Mathematical Word Prob le ms
To bridge the gaps between natural language and mathe-
matical language in solving word problems in Mathematics the
researchers of this study developed an instruction model for the
solution of mathematical word problems.
The two researchers are teachers in two colleges in Israel.
There are many definitions of literacy. The abundance of definitions
derives from the extension of the concept from the wr
itten language
connection and from language altogether. Today literacy represents also
orientation and competence in any area, and thus there are those who
prefer to relate to it in the plural, and to differentiate between different
literacies: linguistic literacy, computer literacy, mathematical literacy,
and so on.
One of them is a teacher of Mathematics and the other is a
teacher in Linguistics. The teacher of Mathematics has encoun-
tered many difficulties of students in solving word problems.
She realized that the difficulties emerged from understanding
the text literally and mathematically. She decided to build a tool
with a teacher in Linguistics to bridge the gaps between Ma-
thematics and Linguistics. The two researchers began inter -
viewing children asking what where the difficulties in solving
word problems. A nine - stages instruction model was built
after the interviews. In several cases, a third party, a colleague
who teaches Mathematics was involved, and took part in the
deliberation process until agreement was reached. After the
model was built it was validated by six experts in Mathematics
and Mathematics education. The nine–stage model was vali-
da te d by 34 student-teachers specializing in mathematics
teaching at a teachers’ seminary in a college in Israel. Only then
it was used on three students: a student from elementary school,
a student from Junior high school and a college student. The
data were taken from word problems in Mathematics, in which
these students were asked to answer in two stages: first they
were asked to answer them without any instruction and then
they were asked to answer them with this instructional model.
Following is our nine-stage instruction model for the solution
of mathematical word problems (schemas appear in the diagram
that follows):
Reading the Problem
The first stage involves reading the problem from the bottom
up, as a way of collecting the details. The action of reading at
this stage is an exposure to the meaning, where the location of
the meaning is in the text. The process of reading is an accu-
mulative process from the smallest units (the words) to the
largest units (the whole text).
Understanding the Linguistic Situation
The next stage involves reading the problem for a second
time. The action of reading at this stage will be called in this
paper the warming up stage: a multi- directional search as a
way of brainstorming. At this stage the reader will ask himself
the following questions:
1. Are all the words clear?
2. Are all the sentences clear?
3. What are the keywords?
4. Do I understand the keywords?
5. What is the question?
6. Do I understand the question?
7. How can I describe the problem in my own words?
Understanding the Mathematical Situation
We define the mathematical situation as the mathematical
context of the problem, which relates to two types of
information: the data and the question. The data relate to all the
expressions that we assume to exist in the problem. They can
appear in explicit form or in implicit form. The explicit data are
those that are mentioned in the text, and the implicit data are
axioms, theorems, and implicit facts that can be used in the
solution of the problem. The question points in the direction of
the expression we want to find.
To understand the mathematical situation there is a need to
examine the data and the question and to have a good grasp of
the subject at hand. It is possible to be assisted, when necessary,
by taking apart, demonstration, exercise, and illustration.
In the first stage it is possible to break up the given problem
into explicit data, implicit data, and a question, and in the
second stage it is possible to demonstrate the problem with
specific instances and to interpret the problem by means of a
picture, a table, a diagram, or a graph, that can help to simplify
the problem.
A problem solver needs to identify the known facts and the
logical-mathematical conditions of the problemthe connec-
tions and relationships between the mathematical data and the
logical analysis of the problem. In other words, the connections
between the classical elementsnouns connected to numerical
quantifiers in different sentences of the text and time-space
relationships between objects or events that appear in the
te xtand the semantic relationship between the classical
elements that are the verbs that appear in different sentences of
the text (Nesher & Katriel, 1977; Hershkovitz & Nesher, 1996).
At this stage the reader will ask himself the following ques-
1. What is my relationship with the mathematical subject of
the problem?
2. Is there a difficulty in the problem?
3. Are all the data clear?
4. Are there implicit data in the problem? (For example: in a
problem that describes a working week, is the intention 7, 6, or
5 working days, as is usual nowadays?)
5. Are there superfluous data? (For example: in a problem
accompanied there superfluous by a background story, are there
superfluous data about the heroes?)
6. Do I understand the connection between the data in the
7. Is it possible to demonstrate the problem in particular
i nstanc es?
Matching the Mathematical Situation to the
Linguistic Situation
This stage involves reading the problem once more, from the
top down. The action
In the course of acquiring schemas the learner meets new
cases and acts on them according to his/her previous schemas
connected to the same matter. The learner expects a particular
If the result marches his/her expectations, then an expansion
of his/her existing schema takes place. If not, then there occurs
a violation that could cause changes in the schema and the
acquisition of a new schema.
At this stage the solver needs to process the literal infor-
mation for the of reading at this stage is the application of
schemas on the text, where the location of the meaning is in the
reader’s knowledge schemas. The process of reading at this
stage is an accumulative process from the combining of mathe-
matical knowledge schemas with the schemas in the text.
A schema is a mental representation characterized by a fixed
internal network of relationships that is created at a high level
of abstraction or generalization and serves as a template that is
used to clarify specific events (Brown & Yule, 1983; Hiebart &
Carpenter, 1992). The schema is an activity pattern (Piaget,
1980) that enables its owner to act under the same conditions in
a consistent manner by habit, and in addition it has a dynamic
characteristic that allows it to expand in new conditions.
Different definitions of schemas by different researchers appear
in Hershkovitz and Nesher (2003).
purpose of changing it into a mathematical exercise or an
algebraic equation while focusing on the syntactic structure and
the semantic structure of the problem. The problem with pro-
cessing the information necessary for solving the word problem
is one of the main difficulties in the solution of mathematical
word problems.
Processing the literal information for the purpose of changing
it into a mathematical exercise or an algebraic equation is done
by understanding the literal clues, that is the words that support
(helpful clues) or the words that deceive (misleading clues) as
clues for choosing the arithmetic operations needed to solve the
problem. For example: the use of the words more or less
(Nesher, Greeno & Reilly, 1982).
At this stage the reader will ask himself the following ques-
1. Do the nouns in the question appear again in a more
general unit? (For example: given apples and later fruits. It is
important that the problem solver understands that apple is a
2. Do the connectives that appear in the question relate to
different mathematical sizes? (For example: if some number is
7 and the product is x what is the multiplying number?)
3. Are there literal clues in the problem that is certain words
that help as a clue for choosing the arithmetic operation
required for solving the problem?
4. Is it possible to demonstrate the problem by means of a
picture, a table, a diagram, or a graph?
Bringing up Ideas for a Solution
The meaning of solving a problem is to find an arrangement
of steps starting from the given situation (in the problem) until
the desired goal is reached, such that each step is derived from
its predecessor by logical operations (acceptable in the context
of the given problem). The process leading to a solution of the
problem is connected to a suitable choice that is a search for a
method, idea, or steps. It may be necessary to analyze the
problem in different ways, to identify the problem before
attempting its solution (Schoenfeld, 1980). To change the
search to systematic, it is necessary to know problem-solving
strategies. There exist general strategies, and strategies specific
to different types of problems.
Usua lly, students are given problems similar to those they
solved in the past. Therefore, according to Polya (1945), there
raises the question: Do you know a problem close to this one?
In most cases there is no difficulty, according to Polya in
bringing up problems that are already solved and are close in
some way to the present problem. As far as using Polya’s
theory, indeed we have not explained it in as much detail as
Polya did, and the significance of his theory is much greater
than presented here, yet from our experience and from case
studies presented in this paper it is possible to see that for
practical purposes the questions appearing in this section are
likely to help the problem solver.
At this stage the learner will ask the following ques- tions:
1. Is the problem unique
2. Have I encountered similar problems?
3. Is it possible to construct a schema for solving the
problem on the basis of past experience?
Screening the Ideas
After raising different ideas for a solution of the problem, it
is necessary to check each one of them, whether it truly helps to
solve the problem. It is necessary to screen them, and to retain
only relevant ideas.
At this stage the learner will ask the following questions:
1. Does the idea help to solve the question?
2. How does the idea help to solve the question?
Building a Mathematical Model
Researchers who are concerned with the process of building
a mathematical model for a phenomenon agree that the mean-
ing of the process is mathematization of a phenomenon (Yeru-
shalmi, 1997) or, according to Ormell’s (1991) version, a ma-
thematical description of the whole phenomenon instead of
checking isolated parameters in the phenomenon. Consequently,
we define mathematical model building as constructing repre-
sentations in mathematical language like an exercise or an equ-
a t ion.
At this stage the learner will ask the following questions:
1. What will I do as a first stage to solving the problem?
2. Do I know how to solve the problem and to build an ap-
propriate mathematical model?
3. What mathematical model should I use to solve the prob-
le m?
The learner will construct a schema that represents the net-
work of connections between his/her previous knowledge and
the schemas in the mathematical text by means of an interaction
with the following operations: defining the problem and com-
prehending the situation it describes; building a mathematical
model of the mathematical principles relevant to the problem;
understanding the relationships and the conditions pertaining to
the problem; and using of the mathematical model.
Finding the Solution
After finding the mathematical model, it is necessary to solve
it to reach the expected solution. It is important to check if this
is a unique solution or if there is another possible solution; all
possible solutions must be found.
At this stage the learner will ask the following questions:
1. Is the solution unique?
2. What are all the possible solutions to the problem?
It is necessary to check that the solution to the problem is
suitable to the problem itself. That is to say, it is necessary to
return to the original problem, to read it again, and to check:
1. Does the solution make sense?
2. Is the solution appropriate to the linguistic situation?
3. Is the solution appropriate to the mathematical situation?
4. Does the mathematical model that I used fit the problem?
This stage is the most important, because many times it
seems as if we have found the solution, but the solution does
not make sense (for example, we got 2.2 people), and so we
need to redo the process from the beginning. It is worthwhile
testing the solution, and checking all the steps that lead from
the data to the solution.
It is important to note that in every word problem it is
necessary to pass through all the stages. However different
learners need to focus on different stages (since some of the
stages are already automatic). During instruction it is necessary
to go over a different stage each time, and to locate the stages
with specific difficulties for different learners and to focus on
them (examples will follow).
Mathematicians focus also on a further stagethe efficiency
stage. They check whether the solution is efficient, whether it is
possible to solve in a different manner, and whether there exists
a shorter method of solution.
A sketch of the teaching and learning model appears in the
following diagram (Figure 1):
Reading the problem
Understanding the linguistic
Matching the mathematical situation to the
linguistic situation
Screening the ideas
Building a mathematical model
The solution
Bringing up ideas for a solution
Returning to the problem to check whether the answer
makes sense, whether it matches the estimation, whether it
fits the linguistic situation.
Figure 1.
Diagram of the teaching and learning model for the solution of mathematical word problems.
Application of the Teaching and Learning Model
To illustrate the model we bring 3 case studies: a student in
the fifth grade, a student in the ninth grade and a college stu-
We attempted to apply the model both to mathematical word
problems accompanied by an authentic background story, and
to mathematical word problems not accompanied by a back-
ground story.
The Case of Yael
Yael is a college student specializing in mathematics teach-
ing at a teachers’ college in the center of the country. She was
given the Students and Professors Problem described above,
and she made the typical mistake, as found in the literature: 6S
= P. After being taught the model, she solved the problem and
got the correct solution: 6P = S. The student wrote down her
working-process according to the instructional model as fol-
1. Reading the problem First of all I read the problem.
2. Understanding the linguistic situation
Next, I read the problem again. I asked myself if it is all clear
(words and sentences), if I understand the question, and how I
could describe the problem in my own words. Once it was clear
to me that I understood all the words in the problem, I marked
the key words: “times greater than”, “the students”, and “the
professors”. I asked myself, what do they want me to do? What
is the P and what is the S? And here is my description of the
problem: The number of students is 6 times greater than the
number of professors.
3. Understanding the mathematical situation
I checked whether I needed to describe the problem with an
equation with letters. I think that it is not a problem that I need
to find an actual result for, but that I have to give an equation.
Then I realized that I have trouble with this sort of problem.
The given information seems to be clear, but in fact you need to
pay attention the concept “6 times”. It is important to analyze
and understand which given is 6 times the other one. Under -
standing the relationship between the two givens is important
for the solution.
4. Matching the mathematical situation to the linguistic
I looked for verbal clues that might help me solve the prob-
lem. In our case, the concept “times greater than” tells us to
multiply by some amount and so that is the arithmetic operation
we need to perform. Here, I started to draw a diagram to bring
the given information into focus:
Students Professors
Their number is 6 times greater
5. Bringing up ideas for a solution
To solve the problem at this stage I have to build a schema.
I’ve met problems like this in the past, and that will help me
build the schema. I met problems with the concept “greater
than”; to see links between a particular number and another.
My ideas for the solution are:
6P = S; 6S = P; 6S > P; 6P > S
6. Screening the ideas
To solve the problem I check whether the ideas I came up
with are relevant to the problem, and whether they help me to
solve it. I think that the inequality 6P > S does not make sense
because the number of students is 6 times bigger than the num-
ber of professors. The inequality 6S > P makes sense, but does
not solve the problem, because I multiplied the number of stu-
dents by 6, and then made it even bigger. From the information
given in the problem I know that S > P. That is, I did not add
anything that helps me to solve the problem. For that reason I
also discard the solution 6S = P.
7. Building a mathematical model
As I said in the previous stage while screening the ideas, I
need to understand the situation - the correct representation of
the problem - and to see whether I understood correctly the
connections between the given pieces of information. In the
previous stage I ruled out all the possibilities: 6S = P; 6S > P;
6P > S. Therefore the only solution that remained is
8. Finding the solution
In this case there is no numerical result, so the solution is 6P
= S or in another form it could be written S: 6 = P.
9. Control
I chose to read the problem again, and again I noticed the
fact that the number of students is 6 times bigger than the
number of professors. It seems to me that my solution makes
sense. In addition, I substituted numbers and checked myself,
since I know that I have trouble with this type of problem and
sometimes I do the opposite.
The Case of Shiri
Shiri is a good student in the ninth grade. She, however, had
some difficulties in understanding and solving the following
word problem that did not describe every day event and was not
accompanied by an authentic background story. Shiri, was giv-
en the following problem in a school te st:
Find the equation of the straight line, parallel to the line 3x
7y = 4, passing through the point (0, 10).
In the school test Shiri answer incorrect to this problem. We
interviewed Shiri and in the interview she claimed that she did
not know how to solve this problem: “I do not understand what
is written in the problem, so …either I do something or I do not
do anything. Therefore, in the test I wrote y = 3x + 10 and that
was not the correct answer. Shiri explained that she took the
3x from the original equation and added 10 because the line
passes through the point (0, 10)).
To enable Shiri to solve the problem we used all the steps of
the above model.
1. Reading the problemAt the first stage Shiri was asked
to read the problem aloud.
2. Unde rstanding the linguistic situationAt this stage,
once it was clear that Shiri understood all the words in the
problem, we asked her to mark the keywords. Shiri marked
them as follows: “equation of the straight line”, “parallel”, and
“passing through the point” .
3. Understanding the mathematical situationDespite the
fact that Shiri understood all the words linguistically and
marked the keywords, she did not yet understand the mathe-
matical context. Shiri was asked to express what she thought
the problem was about. She said, “To find a new line”. Shiri
was asked whether the given information was clear and whether
she understood the connections between the pieces of informa-
tion and the problem. Shiri did not understand the connection
between the given information and the problem, despite the fact
that the general conceptual frame of the problem was clear to
her she knew what a straight line and a parallel line are.
4. Matching the mathematical situation to the linguistic
situation At this stage Shiri needed to process the verbal
information to turn it into a mathematical exercise (equation).
Shiri was asked a general question about straight lines, “What
is the equation of a straight line?” Shiri answered and wrote
down y = mx + n, but she claimed that in the question there was
no equation of a straight line. Shiri was asked what she though
“parallel” meant, and she answered, “A line that has the same
(slope) m”. At this stage integration took place between Shiri’s
schemas concerning the equation of a straight line and her
schemas concerning the text. Shiri was asked to look at the
equation written in the problem and to think how it might be
possible to convert it into the equation of a straight line like the
one she just wrote. Then Shiri said, “ I need to convert the equ-
ation into the equation of a straight line. Oy! In the test I made
a mistake. In the test I took the coefficient of x, the 3, and I
related to it as if it were the slope and that is wrong. It is for-
bidden to relate to the coefficient of x as if it were the slope,
like I did in the test”.
5. Bringing up ideas for a solution Shiri suggested mov-
ing the 3x to the right-hand side and wrote –7y = 4
3x. Shiri
did not know what to do, and said, “This is still not the familiar
equation of a straight line. I do not know what to do with the
minus”. Then Shiri was asked, “On the basis of your past ex-
perience, can you convert this equation to the equation of a
straight line?” Shiri said, “In fact I can move the 7y to the
other side’ and then I would not have the problem of the minus”.
6. Screening the ideas Shiri was asked, “Which idea
would you chose to solve the problem?” Shiri showed the equa-
tion 7y = 3x - 4 and claimed that it reminds her of the equation
y = mx + n because there is no minus before the y.
7. Building a mathematical model Shiri was asked,
“What can you do now to bring the equation to the same form
as the equation y = mx + n?” Shiri looked again at the equation
and said that in her opinion she needs to “get rid of” the 7.
8. Finding the solutionAfter Shiri’s insight concerning
the equation of a straight line and the meaning of a line parallel
to the given line, Shiri solved the problem correctly. Shiri was
asked whether this was the only solution. She answered, “Since
this is a line parallel to a given line and passing through a cer-
tain point there is only one solution.”
9. Control Shiri was asked whether the solution made
sense and met the conditions of the problem. She answered, “I
think so.” When asked how she could check it, she substituted
the point (0, 10) into the equation of the line and said, “I got a
true statement, and the line is parallel to the given line, so my
s olution is correct.”
The Case of Sivan
Sivan is a 6th grade student, who had some difficulties in un-
derstanding and solving word problems. She said: “I cannot
understand word problems. They have too many words in them
and I do not understand what I have to do”.
Sivan was given the following problem and was asked to
solve it:
Tom has a bottle containing three quarters liter of milk. He
drank one third of the milk. How much did he drink?
Sivan did not understand the problem and she said: I do not
know what to do. To enable Sivan to solve the problem we
used all the steps of the above model.
1. Reading the problem First of all Sivan read the prob-
le m.
2. Understanding the linguistic situation Next, she read
the problem again, and she found the key words: “drank” “con-
taining” “three quarters liter of milk” “one third of the milk”.
3. Understanding the mathematical situationSivan
pointed out that she needed to pay attention to the two givens:
“three quarters of” and “one third of”. Sivan said that she does-
n’t like fractions.
4. Matching the mathematical situation to the linguistic
situation Sivan was asked if there are clues that might help
her to solve the problem. She said: I think that one third of the
milk tells me to multiply or maybe to subtract but I don't know
what to do”.
5. Bringing up ideas for a solution To solve the problem
at this stage Sivan was asked if she met similar problems. She
said that she had met problems like this in the past but only
with integer numbers. We asked her to create a problem with
integer numbers. She created a problem, as follows: Tom has a
bottle containing six liters of milk. He drank one third of the
milk. How much did he drink?Sivan's ideas for the solution of
this problem were: 6 X1/3 or 6-1/3.
6. Screening the ideas To solve the problem Sivan
checked whether the ideas she came up to the problem that she
created were relevant to the problem, and whether they helped
her to solve it. She decided to implement her ideas for her
problem to our problem and she said: my ideas for the prob-
lem are: 3/4 X1/3 or 3/4 -1/3”.
7. Building a mathematical model Sivan did not know
which idea is right and therefore she decided to draw a picture
of the problem. Sivan decided that the subtraction did not make
sense because of the picture that she drew. She said: I can see
in the picture what it means 1/3 of 3 /4”. Sivan built the ma-
thematical model: 3/4 × 1/3 (Figure 2).
8. Finding the solution Sivan found the solution: Tom
drank ¼ liter of milk. She said that she is certain that this is the
only solution because of her drawing.
9. Control Sivan read the problem again and said: The
solution is logical, because of the drawing. In the drawing it is
very clear that if you divide 3/4 to 3 you get 1/4”.
C oncl usi on
In this paper we have tried to show how it is possible to
Figure 2.
The mathematical model 3/4 × 1/3.
bridge gaps between natural language and mathematical lan-
guage in solving mathematical problems by means of an in-
structional model, whereby the addressee processes the text
cognitively. The process of dealing with the verbal text of the
mathematical problem is multi-staged, and necessitates the
implementation of a number of cognitive actions: interpreting
symbols and graphs, understanding the substance, understand-
ing the linguistic situation, finding a mathematical model, and
matching between the linguistic situation and the appropriate
mathematical model.
This nine-stage instruction and learning model transforms
into a complex thought process when fully understood and in-
ternalized. It is a meta-cognitive process that contributes to
students’ conceptualization (Natstasi & Clements, 1990). Know-
ledge of meta-cognitive processes assists in problem solving
and improves the ability to achieve goals.
We recommend teachers to “hitch” mathematical language to
natural language: to avoid problems that have no connection
with reality, to avoid ambiguous problems, to explain to the
students the differences between natural and mathematical
languages and the possibility of combining them.
We recommend teachers to make intelligent use of the in-
struction and learning model suggested above, that is, to adapt
the model and its various stages both to different populations of
students and to the nature of the problems and their complexity.
The instruction and learning model suggested here is suitable
for students in the upper grades of elementary school, in middle
and high schools, and in teacher training colleges. In elemen -
tary school, the majority of problems available to students have
a numerical solution with real life meaning, and so it is im-
portant to understand the situation described in them. In their
continued studies in middle and high school students will have
to cope with problems that do not necessarily have a numerical
solution, and will have to use algebra to solve them.
Graduated work on solution methods of word problems using
schemas built in previous work on simply word problem
solving will enable students to cope with more complex prob-
lems. Moreover, adapting the model for different students will
enable teachers to help every student according to his/her needs
and to pinpoint the focus of difficulty for each student.
Intelligent use of the suggested model of instruction and
learning will also help student teachers, both in their training
and in their teaching practice. Understanding the model will
allow the starting teacher to understand that meta-linguistic
awareness, syntactic and semantic awareness, and awareness of
mathematical schemas are necessary for solving problems in
mathematics. Furthermore, the way the problem is worded and
its correspondence to reality can significantly affect students’
ability to solve the problem.
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