PERFORMANCE AND UNDERSTANDING:
A CLOSER LOOK AT COMPARISON WORD PROBLEMS
Ibby Mekhmandarov Ruth Meron
The Center for Educational Technology
Irit Peled
University of Haifa
This work deals with second graders performance and
understanding in solving compare problems. Children
are asked to solve all six types of compare problems and
explain their solutions. A large proportion of children
who solve a given problem correctly give incorrect
explanations. On the other hand, a large proportion of
children who give an incorrect solution exhibit a partial
understanding of comparison situations. Additional
information about their knowledge is obtained from their
solutions in context free situations.
INTRODUCTION
Following a large body of research on additive word problems, Riley,
Greeno, and Heller (1983) in their work extended by Nesher, Greeno,
and Riley (1982) suggest a psychological developmental theory for
knowledge related to word problems. The theory was built to explain
why some problems can be solved by very young children, while others
can be solved only at a later age.
According to their analysis children can solve additive comparison
problems (termed: ‘compare problems’), which ask about the difference
or about the compared group at the part-part-whole stage. Compare
problems which ask about the reference set require a higher stage,
specifically, the child has to perceive the order relation as a twodirectional inequality.
The purpose of our research is to make a more detailed description of
the child’s knowledge. Children’s performance in all six types of
compare problems is observed and the children are asked to explain their
answers. As a result, it is possible to say more about those who fail and
to investigate whether a child who gives a correct answer really
understands the problem’s structure.
PROCEDURE
Second grade children from 2 classes (n=38) were asked to solve all
six types of additive comparison problems. First, each child solved six
compare problems, one of each type. Later, the children were
individually interviewed. During the interview they were asked to solve
problems that involved non-contextual situations describing relations
between sets, which correspond to the six compare problems. Then they
solved each of the written (contextual) problems again and were asked to
explain the solution process.
RESULTS
For each problem the following values were calculated:
The proportion of correct answers, the proportion of correct explanation
for the correct and incorrect answers, the proportion of correct answers
for context free situations within each of the subgroups (performance ,
explanation). In the presentation we will detail the three dimensional
data table. In this paper only a part of the existing data is presented.
Table 1 details the proportion of correct answers for each of the six
compare problems (C1 - C6) together with the proportion of correct
explanations in this subgroup.
Table 1
Percentages of correct answers and correct explanations for all six types
of compare problems.
___________________________________
Problem
Correct
Correct
type
performance explanation
___________________________________
C1
63
39
C2
73
52
C3
63
27
C4
62
22
C5
39
18
C6
63
21
___________________________________
Note: The percentage of correct explanations for a given problem is
calculated for the subgroup of correct answers to this problem. However,
the percentages in each column are of the total number of children who
answered a given problem.
For each of the six problems presented again during the interview, an
analysis of the different explanations has been made. The explanations
have been categorized according to their content and an effort has been
made to identify the developmental level of each answer on the range
suggested by Nesher et al (1982). The developmental level has not
always been relevant and therefore also not always determined. This
happened when children did not exhibit any effort to construct an image
of the situation. For example, some of them turned immediately to a
verbal cue and used it to decide which direct operation to use.
The following answers are examples of explanations given for
problem C5 (compare 5).
The problem: Dan has 5 books.
Dan has 3 books more than John.
How many books does John have?
Answer 1 (a correct answer): 5-3=2
Correct explanation: Dan has more books and John has 3 books less
than Dan, so John has 2 books.
Incorrect explanations:
1. Dan had 5 books and now he has 3 books. This means that he gave
John 2 books. So John has 2 books.
2. Dan has 5 books. Dan has 3 books. You subtract to find by how
much 5 is more than 3.
3. You subtract because 3 is less than 5.
Answer 2 (an incorrect answer): 5+3=8
Incorrect explanations:
1. Dan has 5. Dan has 3. Together he has 8.
2. John has 3 more than Dan, so John has 8.
3. I added because it says ‘more’.
4. I added because you always add.
Answer 3 (an incorrect answer): John has 10.
Incorrect explanation:
1. Dan has 5. Dan has 3. Together he has 8.
John has more than Dan. He might have 10.
Answer 4 (an incorrect answer): You can’t solve it.
Incorrect explanations:
1a. Dan has 5. Dan has 3. Maybe it’s another Dan.
You can’t tell how much John has.
1b. Dan has 5. Dan has 3. They want to confuse me.
You can’t tell how much John has.
Although many of the explanations are incorrect and involve the
transformation of a compare 5 problem into a simpler problem, still the
type of invented problem and its solution indicates, in some cases that the
child has some understanding of a comparison situation. For example, a
child who gave the second incorrect explanation for answer 1 shows that
she knows how to compare two given amounts. A child who gave the
second incorrect explanation for answer 2 shows that she can solve
compare 3 problems, where one has to calculate the compared set. These
two children perform at level 3 (part-part-whole).
Additional information about the child’s knowledge is deduced from
the performance in context free problems. The context free problems
deal with the relations between sets in a way that corresponds to the six
compare problems. For example, the child is asked to build a set of
objects which has a certain (given) number of objects more than the
number of objects of another (given) set. This request is a context free
situation which corresponds to a compare 3 problem. These situations
involve knowledge which can be considered a prerequisite for
performing the corresponding compare problems. Table 2 shows the
percentage of children who performed correctly in the context free
situations although they did not give a correct explanation.
Table 2
Percentages of incorrect explanations and correct context free
performance for each of the six compare problems.
_______________________________________
Problem
Incorrect
Correct
type
explanation context free
_______________________________________
C1
60
33
C2
48
21
C3
72
26
C4
78
44
C5
81
18
C6
79
9
_______________________________________
Note: The correct context free responses in this table are identified
within the set of children who gave an incorrect explanation. The
percentages are calculated as a proportion of the total number of children
who answered a given problem.
The details of these context free situations together with the
specification of examples of children’s performance will be further
elaborated in the presentation.
DISCUSSION
The findings lead to several observations with regard to the
comparison of the child’s performance with her explanation (taken to
indicate amount of understanding), and with regard to the comparison of
the child’s ability to handle context free situation with her ability to
understand a given problem. The main points are:
1. A large proportion of students who supposedly give a correct answer,
have arrived at this answer by using an incorrect analysis of the situation.
2. Children who give an incorrect answer might have a partial
understanding of the comparison situation.
3. Some children can analyze the set structure in a given problem type
correctly as long as the problem involves context free set relations.
Asking the child to elaborate on the way she solves a given problem
enables us to observe two steps in the process: a. The way the child
perceives the problem. b. The way the perceived problem is solved.
Verschaffel (1994) investigates the problems’ encoding stage by
asking children to retell the problems. He deals with four of the six
compare problems in which the unknown is one of the two compared
sets, as these problems are relevant for checking the consistency model.
According to the consistency model the child expects, after being told
about the quantity of one set, to hear how the other set relates to it.
The children in Verschaffel’s study are fifth graders. Still, many of them
convert an inconsistent compare problem (compare 5 or compare 6) into
a consistent problem (usually compare 3 or compare 4), sometimes
making a correct and sometimes an incorrect conversion.
The children in our study are much younger (second graders),
therefore it is not surprising that they convert a given compare problem
into a non-compare problem. Sometimes the conversion is made into a
change problem, and sometimes into a simpler (even trivial) problem.
The interview enables us to detail the different kinds of problems into
which a given problem is converted. It also enables us to see how the
problem is then handled. These observations give us more information
about the child’s understanding of the different situations. For example,
a child might incorrectly convert a compare 5 problem into a compare 3
problem (keeping the word “more” instead of switching to “less” to get a
correct conversion into a consistent problem). However, this child might
then solve the new compare 3 problem correctly, showing that she has a
partial understanding of comparison situations, and also indicating that
she is able of performing a task which, according to Nesher et al (1982)
requires that the child is at level 3 (holding a part/part/whole schema).
It is interesting to note that a child that has answered a given problem
correctly might, in fact, know less about comparison situations than a
child who answers the problem incorrectly. These findings support the
claim that children’s performance should not be judged in
correct/incorrect terms, and show that even incorrect performance can
tell us a lot about what the child does know.
REFERENCES
Nesher, P., Greeno, J. G., & Riley, M. S. (1982). The development of
semantic categories for addition and subtraction. Educational Studies in
Mathematics, 13, 373-394.
Riley, M. S., Greeno, J. G., & Heller, J. I. (1983). Development of
children’s problem-solving ability in arithmetic. In H. Ginsburg (Ed.),
The Development of Mathematical Thinking (pp. 153-196). New York:
Academic Press.
Verschaffel, L. (1994). Using retelling data to study elementary school
children’s representations and solutions of compare problems. Journal
for Research in Mathematics Education, 25, 2, 141-165.