79
Word Problems:
A Framework for Understanding, Analysis and Teaching
Jarmila Novotná, Leo Rogers
1. Word problems
Throughout the history of mathematics, mathematical problems serve to
carry information, to practice techniques, to teach, and to diagnose the
acquisition of skills. We may be presented with a diagram, or some data,
but until we know what to do with this, it means little. We need some
words, some description, some challenge, in order to know how to
proceed further. In the development of mathematics, semi-literate
cultures needed to know certain arithmetical techniques in order to
function in their economy, and the earliest way of transmitting this
information was orally. Over time, oral transmission developed these
sets of instructions into stories or puzzles, and so the tradition of the
Word Problem was born.
Word Problems then, can be regarded as linguistic descriptions of
problem situations where questions are raised, and the answer obtained
by the application of mathematical operations to numerical or logical
data available in the problem statement. Typically, word problems take
the form of brief texts describing the essentials of some situation where
some quantities or relations are explicitly given and others are not, and
where the solver is required to give an answer to a specific question by
using the quantities given in the text and the mathematical relationships
inferred between those quantities.
“Most life situations are described in words. Word problems
constitute one of the few school mathematics domains which
require mathematisation of situations described in words and the
transformation of a mathematical solution back to the context of
the problem.” (Novotná, 2000).
However, in some cases, in order just to practice the particular
mathematical techniques, the problems are so decontextualised, as to be
virtually meaningless in a real ‘real-life’ situation.
80
For example:
Word problem: To make a blouse, mother needs 1.5 m of 140 cm wide
cloth. How much of 90 cm wide cloth does she need?
This can be solved easily by ensuring the data are all in the same
units, setting up an expression about equivalent areas, and performing a
simple transformation. However, if anyone has experience of working
with cloth, or cutting patterns, they will see how far this is removed from
reality.
Verbally stated numerical problems are not considered as word
problems:
Solve the quadratic equation x2 + 3x – 7 = 0.
This is just an instruction – do this!. The mathematical relationship
given implies certain ways of working, and clearly the solver has to
know how to proceed.
In tackling word problems as with any other problems, the help the
teacher gives is most important.
“arithmetic requires an immediate search for a solution, on the
contrary algebra postpones the search for a solution and begins with
a formal transpositioning from the domain of natural language to a
specific system of representation.” (ELTMAPS ArAl Project p.10)
However, in searching for a solution, the representation system is
very important. The choice of a representation system should be open –
pupils can be supported and encouraged to build their own
representations; lists, pictures, diagrams, which can be examined for a
helpful structure and which may be used if they are found to be helpful
and developable. It is important to encourage the careful transposition
from the natural language in which problems are posed into the formal
arithmetic or algebraic language where the relations between the data are
established.
If we are aware of the various aspects of understanding and solving
word problems this should help us to diagnose the obstacles that pupils
face when trying to solve them. When we know more about how
understanding increases during the stages of the solution process, we
81
may be able to prepare more effective instructional programmes for
word problems.
The following notes are intended as a guide planning lessons, and as
an ‘aide memoire’ for the various possible aspects of the process of
problem solving and the analysis of pupils’ work.
2. Understanding and word problems
Understanding is a very complex process. Much of understanding relies
on cultural contexts, and problems which are expressed almost entirely
in words can be a barrier to some people. Different kinds of
understanding have been proposed to see if we can identify what
happens when an act of understanding takes place. For example:

Instrumental understanding is the ability to apply an appropriate
remembered rule to the solution of a problem without knowing why
the rule works.

Structural (relational) understanding is the ability to deduce specific
rules or procedures from more general mathematical relationships.

Intuitive understanding is the ability to solve a problem without
prior analysis of the problem.

Formal understanding is the ability to connect mathematical
symbolism and notation with relevant mathematical ideas and to
combine these ideas into chains of logical reasoning. (Herscovics &
Bergeron, 1983)
The claim is that we use Instrumental understanding most of the
time. We remember algorithms and apply them to problems. However,
when the algorithm does not work, we get stuck - unless we have some
Relational understanding of the situation which helps us to build a new
solution process from the perceived relations between the data and the
mathematics we already know. With Intuitive understanding, sometimes
we are able to see a solution immediately without being aware of any
internal thinking process, while we have Formal understanding when we
are able to represent the problem in a mathematical language.
82
3. Representations
The use of representations is strongly influenced by a teacher's demands
and pupils' habits. The representation can have different forms, from the
detailed rewriting of the assignment to a more clearly organised form
(word representation) on one hand and a considerably shortened record
such as a draft (graphic representation) on the other. The choice of an
appropriate representation can help to get insight into the problem
structure during the solving process.
Functions of representations




to re-present: Its purpose is to create an adequate mental picture.
This is another way of presenting the problem containing the data,
conditions and unknowns. It can range from a nearly realistic to a
completely schematic drawing, or a mathematical statement.
to organise: Its purpose is to bring order into a solver's already
existing mental picture and knowledge in order to connect them
together.
to interpret: Its purpose is to facilitate the understanding of what are
the unknowns of the problem and to reveal the relationships among
the facts given. It helps to eliminate the formation of erroneous
mental pictures.
to transform: Its purpose is to influence the solver's information
processing by changing the used reference language to another one
that is more suitable for the respective solver and to help
systematically to recall helpful information stored in his/her
memory.
4. The grasping process
The first stage in the solving process is to understand the information
given in the text of its assignment. We will call this process grasping
process, and it is intimately linked with the solvers ability to make an
adequate representation of the problem.
83
Basic stages of the grasping process:
a) identifying separate pieces of information during the reading of the
problem,
b) determining what the question was asking,
c) searching for a unifying view,
d) looking for and finding relationships relevant to the solving process,
e) getting an overall insight (finding how all the pieces of information
are mutually connected).
We can propose three components characterising the levels of a
student’s understanding of word problem assignment:
1. reaching the grasping process stages (levels of understanding of the
assignment are related to the successfully finished stages of the
grasping process; the stages are not necessarily expressed in a
student’s solution in an explicit way, they may also be hidden in
implicit steps)
2. how many times the solver refers back to the assignment (a scalar
component)
3. quality of the grasping process (terminology related to the
terminology used in (Hejný, 1995)):

grasping with understanding if a particular stage of the grasping
process results in understanding what was searched for

incomplete understanding if the solver only grasps a part of the
assigned information

prothetic grasping if no understanding occurs in a particular
stage of the grasping process
5. Solving strategies and their relation to understanding the problem
The level of the solver’s understanding of the word problem structure
strongly influences their solving strategy.
84
Items for the characterisation of the chosen solving strategy:
(i) if the solution was found accidentally or after gaining an insight into
the problem structure,
(ii) if the solution was based on the identification of key words/word
groups in the text or on the insight into the problem structure.
Stages of word problem solving process (Novotná, 1997a)
 encoding stage (grasping the assignment),
 transformation stage (transfer to the language of mathematics)
 calculation stage (mathematical solution of the problem),
 storage stage (transfer of mathematical results back into the context).
Terminology (Novotná, 1999)
 Coding of word problem assignment is the transformation of the
word problem text into a suitable system (reference language) in
which data, conditions and unknowns can be recorded in a more
clearly organised and/or more economical form.
 The reference language contains basic symbols and rules for legend
creation.
 The result of this process is a representation.
 The legend constructed in a pictorial form is called a graphic
representation.
6. Difficulties with solving and analysis of children’s work
“A problem is not truly solved unless the learner understands what
he has done and knows his actions were appropriate.”
“Given the proper understanding of mathematical concepts and
procedures, students would be better able to apply their knowledge
in novel situations.”
(Brownell, 1928)
85

Students’ solving strategies and their relation to students’
understanding of the solved word problems
The level of the solver’s understanding of the word problem structure
strongly influences his/her solving strategy.
Items for the characterisation of the chosen solving strategy:
(i)
if the solution was found accidentally or after gaining an insight
into the problem structure
(ii)
if the solution was based on the identification of key words/word
groups in the text or on the insight into the problem structure
(i) Solution found accidentally
Ota (boy, 15 years old)
Problem: Ota and Pavel each had some money but Ota had
10 CZK more than Pavel. Pavel managed to double his amount of
money and Ota got 20 CZK more. They now found that both of
them had the same amount. How many crowns did each of them
have at the beginning?
I ‘ gessed’.
86
(i) Solution found after systematic trials
Hana (girl, 11 years old)
Problem: Petr, David and Jirka play marbles. They have 198 marbles
altogether. Petr has 6 times more marbles than David and Jirka has 2
times more marbles than David. How many marbles has each boy got?
(ii) Identification of key words in the assignment
Filip (boy, 12 years old)
Problem: Petr, David and Jirka play marbles. They have 198 marbles
altogether. Petr has 6 times more marbles than David and 3 times more
marbles than Jirka. How many marbles has each boy got?
87
(ii) Understanding of the problem structure
Jana (girl, 12 years old)
Problem: Petr, David and Jirka play marbles. They have 198 marbles
altogether. Petr has 6 times more marbles than David and Jirka has
2 times more marbles than David. How many marbles has each boy got?
 Classification of solving strategies
The main classification of students' strategies is based on their division
into arithmetical and algebraic ones. But some students use such
strategies which do not fit any of these types and sometimes it is even
impossible to determine the strategy.
Arithmetical strategies
 The student solves the problem using only arithmetical means.
 The student grasps the structure of the problem. (Solutions in which
estimations, visualization etc. are used also belong to this subgroup.)
Algebraic strategies
Solutions where one or more equations are used.
88
(Jitka, girl, 12 years old) – Arithmetical strategy
Problem: The total fee which Mr. Novák’s daughters, Pavla and Marie,
received was 181 CZK. Marie had 37 CZK more than Pavla. How many
crowns did each daughter receive?
Marie got 109 CZK.
Pavla got 72 CZK.
(Zdeněk, boy, 14 years old) – Algebraic strategy
Problem: A buffet sells three different dishes – pizzas, hamburgers and
langoses1). In one day 288 dishes of hamburgers and langoses were sold
altogether. Four times more pizzas than hamburgers and seven times
more langoses than hamburgers were sold. How many dishes of each
kind were sold?
Pizzas
Hamburgers
Langoses
Hamburgers and Langoses
Pizzas were 9, hamburgers were 36 and Langoses 252 pieces.
89
Unidentifiable strategies
It is not possible to determine the strategy used.
 The role of past experience (Novotná, 1997b)
Positive transfer effect: The performance on a current problem benefits
from previous problem solving
Two ways how the solver can use his/her past experience of previous
problems: Ignoring the previous problem, using the previous problem.
(Martin, boy, 13 years old)
Problem a: Petr, David and Jirka play marbles. They have 198 marbles
altogether. Petr has 6 times more marbles than David and Jirka has
2 times more marbles than David. How many marbles has each boy got?
Problem b: Petr, David and Jirka play marbles. They have 198 marbles
altogether. Petr has 6 times more marbles than David and 3 times more
marbles than Jirka. How many marbles has each boy got?19
Problem a
Altogether 198
David 1x more than
David.
Jirka 2x more than David.
Petr 6x more than David.
David has 22 m.
Jirka has 44 m.
Petr has 132 m.
Problem b
alt. 198
David 1x more than
David.
Jirka 2x more than David.
Petr 6x more than David and 3x more than
19
Problems were used in the given order. The Problem 2 structure is more
complicated for solvers.
90
Use of letters
Levels of dealing with letters during the period of grasping the meaning
of the assignment (encoding stage), and the period of mathematization of
the solving process (transformation and calculation stages)

Situation 1 – The assignment does not contain algebraic elements
(Novotná, 1997a):
Four stages of the transition from an arithmetical to an algebraic way of
using letters in the written record of assigned information:
1a) Solvers use one letter for labelling several values, the letter is a
symbol of a general unknown for them.
1b) Solvers use one or more letters in the encoding stage without
working with them in the transformation stage, the unknown is only
used as a label for something that is to be found.
1c) Solvers consciously use letters for labelling required values and for
describing assigned relationships, arithmetical models are more
important and thus the arithmetical solution is used.
1d) Solvers use letters for labelling the values and algebraic operations
are carried out and solved. The conditions for the successful use of
algebraic methods have already been created.
1a) (Jarda, boy, 12 years old)
Problem: Petr, David and Jirka play marbles. They have 198 marbles
altogether. Petr has 6 times more marbles than David and Jirka has
2 times more marbles than David. How many marbles has each boy got?
marbles . .. 198
Petr …. x 6x more than
David … x
Jirka … x 2x more than
91
1b) (Franta, boy, 12 years old)
Problem: Slávek has by 27 stamps more than Honza, Jakub has š times
more of them than Slávek. If Slávek has 138 stamps, how many stamps
have all of them altogether?
SL. 138
J. 3 times more
H. by 27 less than
X = altogether
Altogether they have 663 stamps.
1c) (Marta, girl, 15 years old)
Problem: In a packhouse a blend of coffee for 240 CZK/kg is being
prepared. How can be prepares 35 kg of the blend, if two types of coffee
are available, one for 200 CZK/kg and one for 280 CZK/kg?
92

Situation 2 – The assignment contains algebraic elements
(Novotná – Kubínová, 2001)
Four stages of dealing with the assignment:
2a) Solvers ignore data which are not assigned as concrete numbers,
their ability to work with algebraic representations is not developed.
2b) Solvers use letters only as labels for something that is to be found by
calculations, the value assigned by a letter is handled as an
unknown.
2c) Solvers are aware of the nature of data assigned as letters; by
substituting a concrete number a letter, they change the problem into
a pure arithmetical one. The symbolic algebraic description of the
situation is not yet fixed in their knowledge structure.
2d) Solvers are able to work successfully with data assigned in both
arithmetical and algebraic languages.
Examples - Age of solvers: 13 years
Problem: A packing case full of ceramic vases was delivered to a shop.
In the case there were 8 boxes, each of the boxes contained 6 smaller
boxes with 5 presentation packs in each of the smaller boxes, each
presentation pack contained 4 parcels and in each parcel there were
v vases. How many vases were there altogether in the packing case?
2a)
4 parcels:
packs:
sm. boxes:
boxes:
4.1=4 parcels
5.4=20 packs
20.6=120 sm. boxes
120.8=960 packs
In the case are 960 articles
93
2b)
1 case
8 boxes in
6 smaller boxes in
5 packs in
4 parcels in
? v vases in
In the case are 960 vases.
2c)
8 boxes in 1 are 6 smaller boxes 1. 5 packs 1. 4 p
v=6
In the case are 3520 objects.
2d)
8 . 6 = 48 boxes
48 . 5 = 24 smaller boxes
240 . 4 = 960 parcels
in 1 parcels … v vases
94
7. Programme for examining pupils’ solutions
Ideas for the classroom:
This is a proposal of how to implement the analysis of students’ ways of
grasping the word problem structure and of their solving strategies in a
teacher’s everyday teaching.
 Review and extension of ideas:
Look for solving word problem process stages. Ways of recording of the
assignment – reference languages, mathematical and psychological
views.
 Exploration and discussion:
Consider the classification of various types of information given in the
written records – advantages and disadvantages. The influence of the
chosen way of recording information. Arithmetic and algebraic solving
strategies.
 Extensions/Additions:
What is the role of the teacher and of the student’s previous experiences
 Possible ideas for follow-up:
 Creation of a set of word problems, their use in your own class with
an analysis of students’ solving processes and results
 Finding obstacles and proposing re-educational therapies
 Comparison of students’ work in other domains of school
mathematics or in other subjects with the results of the analysis of
the word problem solving process
 Deep analysis of one (or more) discovered phenomenon.
 Deep analysis of types of word problems and proposed solving
strategies in textbooks
95
References
Brownell, W.A. (1928). The Development of Children’s Number Ideas in
the Primary Grades. The University of Chicago.
Comenius, J.A. (1631). Didacta Magna.
Gavora, P. (1992). Student and Text. Bratislava, SPN. (In Slovak.)
Hejný, M. (1995). Grasping of word problems. Pedagogika, 4, 1995. (In
Czech.)
Herscovics, N. – Bergeron, J.C. (1983). Models of Understanding.
ZDM, 2.
Kubínová, M. - Novotná, J. - Bednarz, N. - Janvier, B. and Totohasina,
A. (1994). Strategies Used by Students when Solving Word
Problems. Publications DMME, 9, Praha.
Novotná, J. (1997a). Phenomena Discovered in the Process of Solving
Word Problems. In: Proceedings ERCME 97. Praha, Prometheus.
Novotná, J. (1997b). Using Geometrical Models and interviews as
Diagnostic Tools to Determine Students‘ Misunderstandings in
mathematics. In: Proceedings SEMT 97. Praha, Prometheus.
Novotná , J. (1998). Cognitive Mechanisms and Word Equations. In:
Beiträge zum Mathematikunterricht 1998, Vorträge auf 32. Tagung
für Didaktik der Mathematik vom 2. bis 6.3.1998 in München. Ed. M.
Neubrand. Hildesheim, Berlin, Verlag Franzbecker p. 34-41.
Novotná, J. (1999). Pictorial Representation as a Means of Grasping
Word Problem Structures. Psychology of Mathematical Education,
12. http://www.ex.ac.uk/~PErnst/
Novotná, J. (2000) Analysis of Word Problem Solutions. Praha, Charles
University (In Czech.)
Novotná, J. – Kubínová, M. (2001). The Influence of Symbolic
Algebraic Descriptions in Word Problem Assignments on Grasping
Processes and on Solving Strategies. In: Proceedings of the 12th ICMI
Study Conference The Future of the Teaching and learning of
Algebra. The University of Melbourne, Australia, Melbourne.
Odvárko, O. et al. (1990). Methods of solving mathematical problems.
Praha, SPN. (In Czech.)
Polya, G. (1962). Mathematical Discovery. John Wiley & Sons.
96
Resnick, L.B. and Ford, W.W. (1981). The Psychology of Mathematics
for Instruction. Lawrence Erlbaum Assoc., Publ.
Skemp, R. (1971). The Psychology of Learning Mathematics. England”
Penguin Books.
Toom, A. (1999). Word Problems: Applications or Mental
Manipulatives. For the learning of Mathematics 19, 1.
Verschaffel, L. - Greer, B. and De Corte, E. (2000). Making Sense of
Word Problems. Sweets & Zeitlinger Publ.
Acknowledgment: The research was partly supported by Projects Grant
Agency of the Czech Republic 406/02/0829: Student oriented mathematical education, and Research Project: MSM 13/98:114100004:
Cultivation of Mathematical Thinking and Education in European
Culture.