A study on the changes in the use of number sense in secondary students

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A study on the changes in the use of number sense in secondary students

Rut Almeida, Alicia Bruno

To cite this version:

Rut Almeida, Alicia Bruno. A study on the changes in the use of number sense in secondary students.

CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Charles University in Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic. pp.238-244.

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number sense in secondary students

Rut Almeida and Alicia Bruno

Universidad de La Laguna, San Cristóbal de La Laguna. Spain, rutalca@gmail.com

Number sense involves the understanding and the abil- ity to use numbers and operations in a reasonable and flexible way. This paper refers to part of a qualitative study that is being conducted to analyse the development of number sense through an intervention designed for this purpose. Our work aims to analyse the strategies used by 8^th grade students when solving numerical tasks before and after an intervention. An initial and a final interview were conducted on 11 students to analyse the effect of the intervention on their tendency to use differ- ent types of strategies. The results show improvements in the understanding of fractions and on the use of number sense, but also underscore the difficulties still encoun- tered by some students.

Keywords: Number sense, rational numbers, secondary education, strategies.

RATIONALE AND PURPOSE

In recent decades mathematics education has focused on the development of skills that make students com- petent to use this knowledge in a flexible and justi- fied way. These kinds of mathematical skills include an understanding of numbers and operations and the ability to use them flexibly, exhibiting different strategies for handling numbers and operations, and the ability to evaluate the reasonableness of results, which is commonly called number sense (McIntosh, Reys, & Reys, 1992). This term appears in the curricula of several countries and is an important topic to be studied. Research intended to assess number sense in students and teachers in primary and secondary education showed a lack of these abilities (Reys & Yang, 1998; Veloo, 2010; Yang, Reys, & Reys, 2009). Although research indicates this is a problem for many students, there is a minority who made use of number sense.

At this point some questions arise: Which of these strategies do they know? Can students learn number

sense strategies through a suitable intervention?

What changes occur after a classroom intervention regarding the use of number sense strategies?

BACKGROUND

The term number sense refers to a broad set of skills and knowledges, and as such it does not have a closed definition. However, some researchers agree that number sense is recognised in the action itself and that it includes the knowledge and flexible use of numbers, operations and their properties, the use of different strategies to solve numerical problems and being able to recognise the reasonableness of the problem statement and data, the way it is solved and the result obtained (McIntosh et al., 1992; Sowder, 1992).

The term number sense is included in the curriculum of different countries, such as Australia, the United States, Canada and Spain, but it seems that this ability is not being sufficiently developed in every country, despite being featured in the teaching objectives.

In an effort to devise a more operational term, there have been attempts to describe it by components (McIntos et al., 1992; NCTM, 1989; Reys & Yang, 1998;

Yang, 2003). They all agree on the aspects involved in number sense, though some are more detailed. For this study the framework considered to characterise number sense is composed by the following components: (1) understand the meaning of numbers; (2) recognise the relative and absolute size of numbers and magnitudes using estimates or numerical properties to make comparisons; (3) use benchmarks to estimate a number or magnitude when comparing or doing cal- culations; (4) use graphical, manipulative or pictorial representations of numbers and operations; (5) understand operations and their properties; (6) understand the relationship between the problem’s context and the operation required; (7) realise that there are

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A study on the changes in the use of number sense in secondary students (Rut Almeida and Alicia Bruno)

239 multiple strategies; (8) recognise the reasonableness

of the problem (data, strategies and results).

Research on number sense has focused on evaluating primary education students and teachers (Veloo, 2010;

Yang, 2003; Yang et al., 2009), with less research being conducted on secondary education (Reys & Yang, 1998; Veloo, 2010). All of these studies have found that there is a lack of number sense in these groups when solving numerical tasks. A tendency to use algorithms and written computation, and a lack of relationship between good written computations and good number sense have been shown in several studies on high grades of primary education and lower secondary education (Reys & Yang, 1998; Veloo, 2010). In contrast, it was revealed that students with higher academic marks in mathematics performed better in number sense tests (Veloo, 2010) as did those with good mental computation and estimation skills (Sowder, 1992). In an effort to address this situation, interventions to develop number sense have been designed. Markovits and Sowder (1994) established that number sense can be developed, but over a long period of time, and that students, in addition to gaining new knowledge, reor- ganised prior knowledge. Those studies that included an intervention to develop number sense agree on the idea that activities must contain a process of numerical exploration, the search of number and operation properties and a methodology based on the discussion (Veloo, 2010; Yang, 2003). All of them have shown that this type of learning is more significant than traditional learning and that it is possible to develop number sense and the activities they used for it. These studies only show pre-test and post-test results that enable us to see that the activities worked, but qualitative data concerning the changes in their use of strategies is still needed.

OBJECTIVES AND METHODOLOGY

This work is part of a broader study whose objective is to identify the strategies used by students before and after an intervention to analyse the possible changes in their use of number sense. Since this term encom- passes multiple concepts that are developed over the course of mandatory schooling, in this case we have opted to focus our study on three of the lesser studied components in research into this topic involving secondary students. The intervention focused on the development of three components from the framework involving whole, decimal and rational numbers: (3)

use of benchmarks; (4) use of graphical representations of numbers and operations; (8) recognise when the result is reasonable. The design was intended to foster these three components, although the use of other components was expected since they are not independent.

The sample consisted of two groups (25 and 22) of 8^th grade students (12–13 years old) from a public secondary school in Spain. These students were involved in an intervention designed to develop and encourage the use of number sense. The students’ existing knowledge of number sense was in keeping with Spain’s curriculum in traditional numerical learning, which at this level involves consolidating students’ knowledge of whole, decimal and rational numbers.

Initial test

The students were given an initial written test that included 12 items designed to evaluate the strategies used when facing numerical tasks focused on the use of the three components mentioned above. This initial test revealed the students’ shortcomings in the use of benchmarks and in making graphs to aid in estimating numbers and in operations, their preferences for using rules or algorithms, and conceptual errors, especially with fractions (arranging them considering differences between the numerator and denominator, making incorrect graphical representations of fractions and operations) (Almeida & Bruno, 2014). The test yielded background information that was used to design the classroom intervention tasks.

Classroom intervention

The classroom intervention took place over the course of eleven 50-minute sessions and was based on a col- laborative environment and on discussions between students of each task. They solved tasks working individually, in pairs, in small groups or as a whole, though always showing the strategies they used so as to discuss them with the rest of their classmates in a final discussion. Since, generally, students tended to use memorized rules, at the end of each task during discussion they were asked to find different strategies in order to explore numbers and find number sense strategies. This methodology allows students to share their knowledge and to discover number sense strategies without the intervention of the teacher/researcher. The students were given written tasks designed by the researchers that were intended to show that an exact answer or calculation is not always necessary

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to resolve certain questions, to highlight the impor- tance of questioning one’s results by evaluating if an answer is reasonable or not, and to develop strategies that involve the use or graphical representations and show the usefulness of taking benchmarks. The tasks themselves involved estimating quantities, sorting numbers and doing operations with whole numbers, decimals and fractions while encouraging debates among the students to find the most suitable strategy.

Interviews before and after the classroom intervention (case studies)

The initial test was also used to select 11 students who were interviewed before and after the classroom intervention with the aim of making a case study of the possible improvement. Both interviews, while designed by the researchers, also relied on previous research (Yang & Huang, 2004; Reys & Barger, 1991).

Their aim was to assess the use of the three components being studied. The interviews consisted of two phases. In the first one the students solved all of the tasks without the researcher’s intervention. Once they had finished, they were asked to explain each answer by trying to find new strategies without using a written computation method. If they had not used a graphical representation, they were encouraged to find a way to use one. The initial interview had six items and the final one had five. Regarding the analysis of the data from interview, the answers of the eleven students were analysed to identify the cor- rectness (1 for correct answers and 0 otherwise) and the type of strategy used. A category system adapted from Yang and colleagues (2009) was used to classify their reasoning: number sense based (NS), when they used exclusively one or several components of the number sense framework; partially number sense based (PNS), if they combined the use of number sense components by using memorised rules and/or algorithms; not number sense (NNS), when they only made use of algorithms or memorised rules; other (Oth), when students do not provide sufficient grounds to identify the reasoning that led them to the final answer(s); blank (B), when the question is unanswered.

The NS and PNS categories also considered the type of component used.

Due to space limitations, in this paper we present the results of one item that was similar in the initial test and in both interviews. This allows us to compare the answers of the eleven students before and after the intervention. Focusing on just one item, we wish to

exemplify the type of analysis that we are undertak- ing. These items were the following: Initial interview,

“Sort from smallest to largest the following numbers:

2/5, 7/8 and 4/3”; Final interview, “Sort from smallest to largest the following numbers: 7/8, 0.3, 4/3, 0.55 and 2/9”. We opted for this problem type involving sorting fractions because of the poor results obtained on the initial test, along with the variety of answers given based on rules. It was also presented in the initial and final interviews, and despite having a standard problem statement, it can be solved using the components of number sense that are being studied, which allows us to observe how the students put them into practice. Specifically, the objective of these items concerning the number sense was to encourage students to use benchmarks (component 3) and/or graphical representations (component 4) to compare fractions without the need to use written computation. All of the tasks in the initial and final interviews were designed such that they could be solved using the components in this study. The initial and final interviews were separated by a period of three months. Over this time the students were involved in a classroom intervention. Of the eleven sessions, two were devoted to working on tasks related to the items analysed, i.e.

sorting rational numbers. An example of one intervention task is shown below.

The students were given a problem to solve individually for later discussion with the whole group. The statement was: “Sort from smallest to largest the following numbers: 9/20, 8/5 and 3/10”. After solving the problem individually, they were asked to present their strategies on the blackboard. In this case the instruc- tor’s intervention was not needed and the students found all of the strategies expected: (1) compute the least common multiple to apply an algorithm and express all the fractions with a common denominator;

(2) apply the division algorithm to express fractions as decimal numbers; (3) graphically represent the fractions; (4) use the benchmarks ½ and 1 to compare the fractions. The idea was to show every possible way they knew to solve the problem so that the students could see the different possible strategies and be able to choose the one they preferred. In those cases with insufficient variety, they were encouraged to explore numbers in different ways such as understanding the meaning of numbers, the magnitude, use of benchmarks, graphical representions, etc.

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241 RESULTS AND DISCUSSION

The eleven students interviewed answered the two items mentioned above, using different methods when asked to do so. Table 1 shows a summary of the results by the category of the strategy the students used for each task. The number in parentheses shows the component from the number sense framework that they primarily used. The first problem (1^st) was the one the students did on their own without the interviewer’s intervention, and the rest resulted from asking them to use new strategies to solve the task.

In the answers we find four out of five different categories: NS, NNS, Oth and B. A description of the strategies included in each category is presented below. The PSN category did not appear in this item, as the students made exclusive use of number sense components or rules independently, though there were strategies that combined the use of both in other items in the interviews.

Number sense (NS)

Component 1: Using the properties of numbers to express fractions as equivalent fractions or decimals as fractions indicates an understanding of the meaning of numbers. An example of this is provided by student S4, who states for the task in the final interview “(...) we can express decimals as fractions to compare them, for example, 0.3 is equal 3/10 and 0.55 is around ½ (...)”.

Component 3: Using benchmarks to facilitate comparisons. The students used different benchmarks, some using more than others to compare against.

One example of the use of benchmarks is provided by student S1 who, after the intervention, used four different benchmarks (1, ½, 1/3 and ¼): “7/8 is almost 1, 0.3 is a third, 4/3 is greater than 1, 0.55 is a half and 2/9 is close to ¼”.

Component 4: Using graphical representations to ascertain a number’s magnitude for comparison purposes, especially for fractions. Figure 1 shows an example from the final interview.

Not number sense (NNS)

We found two kinds of strategies that made use of rules and/or algorithms. The most common in the initial interview was to compute the least common multiple to apply an algorithm and express the fractions with a common denominator. The other strategy

Student Initial interview’s strategies Final interview’s strategies

1^st 2^nd 3^th 1^st 2^nd 3^th 4^th

S1 0NS(1) 1NS(3) 0NS(4) 1NS(3)

S2 1NS(3) 1NS(4) 1NS(3) 1NS(4)

S3 1NNS 1NS(3) 1NS(4) 1NS(3) 1NS(4)

S4 1NNS 1NS(3) 1NS(4) 1NNS 1NS(3) 1NS(1) 1NS(4)

S5 0NNS 1NS (3) 0NS(4) 0NS(3) 0NS (4)

S6 0B 1NS(3) 1NS(4) 1NS(3) 0NS(4)

S7 1NNS 1NS(3) 0NS(4) 1NNS 1NS(3) 0NS(4)

S8 0NNS 0NS(3) 1NS(4) 1NNS 1NS(4)

S9 0NNS 0NS (4) 0NS(3) 0NS(4)

S10 0Oth 0NS (4) 0NS (4)

S11 0B 1NS(3) 0NS(4) 1NS(3) 1NS(4)

0: Correct final answer; 1: Incorrect final answer.

NS: Number sense; NNS: Not number sense; Oth: Other; B: Blank.

(1) Understand the meaning of numbers; (3) Use of benchmarks; (4) Use of graphical representations;

Table 1: Strategies classification for the item selected in both interview by student

Figure 1: Answer from S2 in the final interview

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found in this category was a memorised rule in which they established that a fraction was greater than other if the difference between the numerator and the denominator was smaller, without making sense of the fraction concept and the incorrectness of the rule for cases with different denominators. In the final interview, the most common strategy was to apply the division algorithm to express fractions as decimals.

Other and blank (Oth and B)

Common to the answers in these two categories is the fact that they did not have a justification for the reasoning or, in the case of “Blank”, even an answer.

Analysis of the answers

An individual analysis of the students’ answers re- veals the improvements and/or differences in the use of strategies (Table 1) when solving this item.

NS answers before and after the intervention Students S1, S2 and S6 used number sense strategies as their initial answers in both interviews, though we observed more confidence in the use of number sense strategies after the intervention, since in the initial interview students argued that they were used to use exact written computation so they were not allowed to use what they named as “logical reasoning”

referring to estimates or the use of benchmarks, but in the final interview they made use of number sense strategies without this statement, a hint of changes in the didactic contract during the intervention. S1 used benchmarks and graphical representations correctly in both interviews, whereas students S1 and S6 had problems with the graphical representations of fractions and decimals before and after the instruction.

We regard Student 1 as being representative of this group.

In the initial interview, Student 1 used three different strategies to answer the task, but only one of them was completely correct. In his first answer his reasoning was correct, making use of the fraction’s properties to find equivalent fractions, but he made a mistake when estimating the magnitude of 2/5. The student was unable to graphically represent the fractions although he was able to correctly compare them with 1 and ½. After the intervention, the student decided to use this last strategy but in this case using more benchmarks (1, 1/2, 1/3 and 1/4) before stating that he did not know how to represent fractions. This student, although he knew some number sense strategies, did

not show any apparent improvement involving the use of graphical representation. He demonstrated a mastery of the use of appropriate benchmarks, even more so after the intervention.

NNS answers before and after the intervention Students S4, S7 and S8 were characterised by their preference to use rules for the first answer despite knowing other procedures involving either graphical representation or benchmarks. We regard Student 7 as being representative of this group.

Student 7 had similar results in both interviews; in his first answers he gave rule-based reasoning, although in the initial interview this involved computing the least common multiple and in the final one it was related to the division algorithm. For the other strategies used, he followed the same reasoning in each interview, including the use of benchmarks and graphical representations, but although it was correct, he was unable to interpret it. Therefore, the student made no progress in this task after the intervention.

Improvement in NS after the intervention Students S3, S5 and S11 were characterised by exhibiting a change in strategy between the two interviews toward an improved use of number sense. The case of student 11 is described in greater detail below.

Student 11 used the same strategies in both interviews but with a clear improvement in the second case. Both times she used benchmarks and graphical representations; regarding benchmarks, she exhibited a rich- er justification in the final interview, choosing two benchmarks (1 and ½), while in the initial interview she said she was not able to do so, and giving more accurate explanations of their comparison; as con- cerns graphical representations, she demonstrated an improvement in this aspect as well, being able to represent all the numbers. This last improvement is evident since in the initial interview she was not able to represent fractions correctly, dividing the units into unequal portions.

No improvement in NS after the intervention Students S9 and S10 used incorrect strategies when they attempted to employ improperly memorised rules before the instruction. However, they exhibited a change in intention after the final interview by trying to make sense of fractions and their magnitude instead of using memorized and non-argued

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243 rules. They applied number sense strategies, although

conceptual errors related to fractions that remained after the intervention impeded them from obtaining the correct answer. As an example of this group, let us consider the answers given by student 9.

In the initial interview, Student 9’s answers demonstrated a lack of understanding of fractions; she applied a memorised rule in which she decided the magnitude of the fractions based on the difference between the numerator and denominator, which led her to an incorrect answer. She also tried to use a graphical representation in which she divided the units without realising that the pieces had to be equal for each fraction. After the intervention, she demonstrated an improvement by attempting to make sense of the strategy used and not applying a memorised rule. She tried to apply the use of benchmarks and graphical representations but the improvement was insufficient, given her misconception of the meanings of the numerator and the denominator: namely, she reversed their meanings (Figure 2).

CONCLUSION

Students who used number sense strategies in the initial interview as their first answer continued to use these strategies, but were more confident using them, knowing that they were allowed to use any strategy they knew (S1, S2 and S6). In contrast, after the intervention some students preferred to use strategies classified as not number sense as their first option (S4, S7 and S8). These students expressed a knowledge of other number sense strategies and yet opted to con- tinue with rule-based methods, as they were more at ease with them and because they felt their teachers expected them to use these procedures. There were also students who changed their strategies towards the use of number sense, although in some cases their misconceptions concerning graphical representa-

tions of decimal numbers kept them from obtaining the correct answers (S3, S5 and S11).

As other interventions that aim to encourage the use of number sense have shown, developing these strategies is a long-term process (Markovits & Sowder, 1992).

Evidence of this statement is the insufficient improvement of S9 and S10 related with their mathematical misconceptions concerning rational numbers. The use of number sense requires the use of conceptual ideas that, in this case, involve rational numbers, their meaning and properties. Therefore, in cases where misconceptions arise, more time might be needed to develop number sense strategies. These mistakes are particularly evident as they relate to the concept of fraction and to the graphical representations of fractions as a means for obtaining an answer to a problem. Even though not every student showed op- timal improvement, some changes were noted as a consequence of the intervention. The results of the interviews led us to delve into the different options students used to obtain an answer: in some cases

students used a rule-based strategy, but they were able to use number sense strategies as a second or third option. This flexibility in the use of strategies is one limitation of a written test, since students only show one strategy that conceals the number sense they may possess, perhaps because they think that is what teacher expected or because they feel more confident using written computations.

ACKNOWLEDGEMENT

This research was partially funded by the project EDU2011-29324, Formal and Cognitive Skills Models for Numerical and Algebraic Thinking in Primary and Secondary Education Students and in Pre-service Teachers. Funding was also granted thanks to the ULL by the Canarian Agency

Figure 2: Answer from S9 in the final interview

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for Research, Innovation and Information Society, financed 85% by the European Social Fund.

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