STRATEGIES AND PROCEDURES: WHAT RELATIONSHIP
WITH THE DEVELOPMENT OF NUMBER SENSE OF
STUDENTS?
Elvira Ferreira
Higher School of Education of Torres Novas
Lurdes Serrazina
School of Education, Instituto Politécnico de Lisboa
Abstract. This paper analyses the procedures and strategies used by second graders students in a
problem solving context using addition and subtraction of positive whole numbers under a
classroom teaching experiment. This is a qualitative and interpretative case study, with data
collection through participant observation, interviews and documents, namely, reports of classroom
episodes, tasks and student’s involvement in classroom activities. This paper analyses one of the
students, Daniel. The results suggest that Daniel’s preference for certain mathematical procedures
and strategies depended on the context of the problems, namely the types of situations involved
and the development of some components of number sense.
Keywords: procedures, strategies, number sense, addition and subtraction.
INTRODUCTION
During the last decade, the goals and content of elementary mathematics education
have changed internationally (Kilpatrick, Swafford, & Findell, 2001; Verschaffel,
Greer, & De Corte, 2007). The development of number sense is now an essential
aspect of learning mathematics in the first school years, enabling students to solve
problems involving addition and subtraction with positive whole numbers (McIntosh,
Reys, & Reys, 1992). In the 21st century, “helping children develop number sense is
being considered on a global scale as a key task in mathematics education” (Yang, Li,
& Lin, 2008).
This paper reports part of a study which main aim is to describe and analyse how
students develop their number sense in a problem solving context using addition and
subtraction of positive whole numbers, considering problems of real world addition
and subtraction situations (Fuson, 1992). In particular, understanding the strategies
and procedures they use in solving subtraction problems under a classroom teaching
experiment. In this paper will be analysed the strategies and procedures used by one
student (Daniel), when compared with those described in the literature in the field.
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THEORETICAL FRAMEWORK
What is number sense? This is a question which the answer is not easy to obtain.
Greeno (1991) states that “number sense is a term that requires theoretical analysis
rather than a definition” (p. 170) and he suggests that “it may be more fruitful to view
number sense as a by-product of other learning than as a goal of direct instruction”
(p. 173). Dolk (2009) considers that developing number sense in the class setting
“implies giving students the opportunity to think with numbers and operations,
guiding them in the way they look at numbers, and helping them to construct an
active network of number relationships” (p. 5). Developing number relations also
implies that students see numbers as mathematical objects.
McIntosh et al. (1992) define number sense as “a person’s understanding of number
and operations along with the ability and inclination to use this understanding in
flexible ways to make mathematical judgements and develop useful strategies for
handling numbers and operations. It reflects an inclination to use numbers and
quantitative methods as a mean of communicating, processing and interpreting
information” (p. 3). They propose three strands to number sense: (i) knowledge of
and facility with numbers, (ii) knowledge of and facility with operations and (iii)
applying knowledge of and facility with numbers and operations to computational
setting. This definition encompasses the behaviour defined by other authors as
strategy use, and on the belief that promote strategy flexibility is important for all
children, including younger and mathematically weaker children (Kilpatrick et al.,
2001;Verschaffel et al., 2007; Verschaffel, Greer & Torbeyns, 2006).
Thus, strategies are seen as embedded within number sense. Strategies for solving
particular types of problems are often presented as procedures that are followed in
response to the stimulus problem. For Beishuizen (1997) strategy is the “choice out
of options related to problem structure” and procedure is “the execution of
computational steps related to the numbers in the problem” (p. 127).
The discrepancy between formal and informal computation procedures is currently
seen as an impediment to the initial learning and understanding of mathematics
(Blöte, Klein, & Beishuizen, 2000) as well as a hindrance in the development of
number sense and the use of flexible number operations at the end of primary school
(McIntosh et al., 1992; Treffers, 1991). The study developed by Yang (2003)
demonstrates that students’ number sense can be effectively developed “through
establishing a classroom environment that encourages communication, exploration,
discussion, thinking and reasoning” (p. 132).
Many of the studies of children’s strategies and procedures consider mental
computation methods very important in solving addition and subtraction problems
(Beishuizen, 1993; 1997; Blöte et al., 2000; Buys, 2001; Klein et al., 1998; Torbeyns,
Verschaffel, & Gesquière, 2006; Verschaffel et al., 2007). Such problems can be
solved by three types of procedures: one type is the split method (1010); the second is
the jump method (N10) and the third type is called varying, compensation or short
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jump. In the 1010 procedure numbers are decomposed in tens and ones which are
processed separately and then put back together. The 10s (1010 stepwise) is a 1010
procedure that conceptually can be located between the 1010 and the N10 procedure.
The N10 computation procedure (also the variant of N10C) starts with counting by
tens up or down from the first, unsplit number. The A10 (adding-on) procedure also
starts from the first, unsplit number and goes from there to the next ten. The varying,
compensation or short jump refers to bridging the difference in subtraction problems,
like “71 - 69” in one or two steps instead of subtraction the second number from the
first one (Blöte et al., 2000, p. 222) or 86 - 25 = 85 - 25 + 1 = 60 + 1 = 61 (Torbeyns
et al., 2009, p. 80).
All these three procedures belong to “direct subtraction or indirect addition
strategies” (Torbeyns et al., 2009, p. 80), when solving subtraction problems. Solving
a subtraction problem by indirect addition means “that the problem is solved by
adding on from the subtrahend” (Van den Heuvel-Panhuizen & Treffers, 2009, p.
108) and “direct subtraction means that the subtrahend is subtracted from the
minuend (e,g., 71 - 29 = ?)” (Torbeyns et al., 2009, p. 80).
Apart from the nature of the numbers, Van den Heuvel-Panhuizen & Treffers, (2009)
consider that there is more reason to calculate subtraction as an addition and so make
use of the complement principle. It is important to consider subtraction as taking
away (direct subtraction) and as determining the difference (indirect addition). The
same authors say that “in the subtraction problems the context opened up the indirect
addition strategy” (p.109)
The relationship between subtraction and addition is a big idea that children need to
develop. Eventually, it is important that children know either strategy can be used.
“Children need to understand the connection between addition and subtraction.
Furthermore, they need to understand that comparison and removal contexts can both
involve subtraction” (Fosnot & Dolk, 2001, p. 90). Traditionally teachers have often
told learners that subtraction means “take away”. This is a superficial, trivialized
notion of subtraction, if not erroneous (Fosnot & Dolk, 2001).
METHODOLOGY
The aim of the research is to understand how students develop their number sense in
a problem solving context using addition and subtraction of positive whole numbers.
In particular, understanding the strategies and procedures children used in solving
addition and subtraction problems under a classroom teaching experiment. The
purpose of teaching experiments is to develop theories about both the process of
learning and the means that are designed to support that learning (Gravemeijer &
Cobb, 2006). These authors discussing three phases of conducting a design
experiment, which are: (i) preparing for the experiment, which begins with the
clarification of the mathematical learning goals; (ii) experimenting in the classroom,
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and (iii) conducting retrospective analysis, that are conducted of the entire data set
collected during the experiment. In this case, a sequence of instructional tasks was
elaborated as well as a conjectured learning process that anticipated how students’
thinking and understanding might evolve when the instructional tasks were employed
in the classroom. In addition instructional tasks, we also considered the classroom
culture and proactive role of the teacher. In the classroom culture we accounted the
nature of classroom norms, social and socio-mathematical norms (Yackel & Cobb
1996)
The object of study is a group of four children in elementary school integrated in a
second grade classroom. The data collection was done during the school year
2007/2008 and included (i) participant observation, with reports from several lessons,
one or two each month, during the school year, concentrating on the way children
solve addition and subtraction problems with different structures (the sequence of
instructional tasks mentioned above), (ii) interviews, done by the first author, with the
four children three months after finishing the classroom teaching experiment, which
were audiotaped and transcribed; (iii) written documents, namely, reports of
classroom episodes, tasks and students’ involvement in classroom activities. All the
lessons mentioned at (i) were videotaped.
The principal source of data is participant observation with writing of researcher
reports and collection of documents (Yin, 1989; Patton, 2002), by the first author of
this paper. Each observation lasted at least two hours, and included observing
children solving problems and listening to their explanations (Bogdan & Biklen,
1994; Erickson, 1986; Guba & Lincoln, 1994).
According to the research plan, data analysis began simultaneously with data
collection, in order to identify students’ strategies and procedures and how they were
developing them. Data comprised of videotaped the students engaged in solving
problems, during classroom lessons. The lessons were analysed for sequencing of
problems presented by the teacher, and the strategies and procedures used and
discussed by the students.
In this paper, we describe Daniel’s strategies and procedures for four subtraction
problems during classroom teaching experiment.
RESULTS
During classroom teaching experiment Daniel solved several real world subtraction
situations (take away, complete, compare difference unknown and compare referent
unknown). In this section we present the resolutions of one subtraction problem
compare difference unknown and three subtraction problems compare referent
unknown.
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Problem one (compare difference unknown). In the cinema, Room 1 has 215 seats.
Room 2 has 98 seats. How many more seats does Room 1 have compared to Room 2?
Daniel uses a direct subtraction strategy and he uses N10 computation procedure. He
starts from the first unsplit number and then he takes away 98 (figure 1):
Figure 1: The way Daniel solved problem one
When he explains how he carried out the computations, he says:
Daniel: I did 215 – 15 (from 98 decomposed into 15 + 80 + 3) and I got 200.
After, 200 minus 80 is 120. As I had taken away 95, there was still 3 missing
so 120 – 3 is 117.
Problem two (compare referent unknown). Stamps problem (figure 2)
Portugal’s box has 35 more
stamps than Spain’s box. How
many stamps does Spain’s box
have?
Portugal
82 stamps
Spain
..? stamps
Figure 2: Stamps problem
This was the first problem compare referent unknown during classroom teaching
experiment. Daniel uses a direct subtraction strategy and he uses N10 computation
procedure. He starts from the first unsplit number and after he takes away 32 (figure
3)
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Figure 3: The way Daniel solved problem two
First, he takes away 30 and he gets 52. Then 52 – 2 (from decomposition 5 into 2 +
3), and, finally, he takes away 3 and he gets 47, the answer to the problem.
Problem three (compare referent unknown). 303 students from Eleanor’s school
went to the cinema. These are 45 more students than from John’s school. How many
students from John’s school went to the cinema?
Daniel uses an indirect addition strategy and he uses a mixed computation procedure,
N10/A10. He starts from the first unsplit number and after he adds 245 and he gets
290. He says “I already knew in my head that 45 + 45 is 90” (a basic fact that he had
already automated), and plus 200 is 290. After he adds ten and gets 300, a benchmark
number (A10). Finally, he adds 3 and reaches 303(figure 4).
Figure 4: The way Daniel solved problem three
Problem four (compare referent unknown). For lunch, Peter ate a Big Mac which
has 490 calories and Antonio ate a piece of salmon fish. The Big Mac has 295 more
calories than the fish. How many calories does the fish that Antonio ate have?
This is the last problem with this context during classroom teaching experiment. It is
very interesting because first he uses indirect addition strategy and N10 computation
procedure and after he uses direct subtraction strategy and the same computation
procedure, N10 (figure 5), that is, Daniel says that he can use both strategies for
solving subtraction problem, and this helped him to develop the capacity of solving
addition and subtraction problems and number sense.
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Figure 5: The way Daniel solved problem four
DISCUSSION
The results presented above show that Daniel is able to justify his computations using
appropriate strategies and procedures. Although some students display 1010 in
solving subtraction problems, Daniel never solved subtraction problems using 1010.
It seems that this is related to his mental computation ability, that is, to his flexibility
with numbers and their manipulation. This may be related to understanding of the
meaning of numbers and operations, how he uses the reference numbers and how he
recognises the reasonableness of the results. This understanding also provided that
Daniel invented new own procedures.
The findings of this study also indicate that as time progresses, it will be more likely
students use N10 and this may be related to the way subtraction problems are solved.
Another finding of this study indicates that there is a connection between the use of
indirect addition and the context of the subtraction problems they have to solve. This
seems to happen in problems of complete and compare difference unknown, as the
context of these problems help the students, and in particular Daniel, to understand
the relationship between addition and subtraction and to use addition operation. The
use of additive strategy also helped Daniel to use more efficient procedures, inventing
his own procedures. Also there seems to be a relationship between the chosen
strategy and effectiveness of procedures that Daniel was able to use.
The findings of the large study allow us to say that the students were less flexible
with addition than with subtractions problem procedures. Like noted by Blöte et al.
(2000), the students used the 1010 procedure to solve addition problems, whereas, on
that same occasion they used the N10 procedure to solve subtraction problems Those
findings also present empirical evidence, that the development of strategies and
procedures on multi-digit solving addition and subtraction problems were influenced
by social and socio-mathematical settings that the students were involved in a
classroom environment that encourages communication, exploration, discussion,
thinking and reasoning. These findings are consistent with other studies, namely,
Kilpatrick et al. (2001), Verschaffel, et al. (2007) and Yang (2003).
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