TEACHING NUMBER LINE, FRACTIONS, DECIMALS AND PERCENTAGES AS A WHOLE SYSTEM

Regina Reinup Tallinn University

Fractions and decimals have two different meanings. On the one hand they are rational numbers with the concrete places on the number line. On the other hand they express ratio and relationship, which can also be expressed using percentages. This ambiguousness evokes difficulties in calculation. The pupils are often in a quandary which of these meanings of fractions and decimals they must use in different exercises (or in real life situations). They probably do not understand that the reason of difficulties in these exercises lies in choosing the appropriate meaning. Therefore it is very important to show to pupils the whole system of number line, fractions, decimals and percentages. This article presents an overview of this system, and proposes some examples and exercises which can be used in teaching of this topic in the 6th grade.

INTRODUCTION The teaching and learning of fractions, ratio, and proportionality is a complex process as described by many teachers and researchers (e.g. Moss, 2005; De Corte, Depaepe, Op t Eynde & Verschaffel 2005; Adjiage & Pluvinage, 2007). Fractions and decimals can be taught on the one hand as rational numbers, with concrete location on the number line. So we can speak of the triangle of fractions decimals number line (hereafter FDN). On the other hand fractions and decimals express ratios and proportionalities, and are in this sense closely related to percentages. So we can speak of the triangle of fractions decimals percentages (hereafter FDP) as well. Although the pupils can do mathematical operations with decimals and fractions, they come face to face with difficulties when they must use decimals and fractions in sense of ratios and proportionalities (Moss, 2005). Duval (2006) points out that for pupils it is very difficult to change from one semiotic system to other. It is necessary to show to pupils the whole system of number line fractions decimals percentages, and to illustrate it with examples and exercises from real life (Adjiage & Pluvinage, 2007; De Corte et al. 2005) when and why we are in which triangle and how to change own thinking from one triangle to the other. The purpose of this paper is to describe this whole system (hereafter WS), and to find suitable examples and exercises for better understanding it. In the first part of this paper I present the WS. In the second part I describe the 7th grade pupils choices of transformations, and their skills in FDP. In the third part I introduce examples from real life, and in the last part I discuss principles of the design of exercises linked with the WS.

THE WHOLE SYSTEM Fractions Decimals Number Line The pupils first contact with FDN takes place already in primary school. The number line is known from first grades as the line where at first are situated only natural numbers, later whole numbers, decimals and fractions as well. It is very natural that mathematics operations related with number line are first and foremost adding and subtraction, because these operations can be visualized easily on number line, and they can be handled as going forwards or backwards on it. To visualize multiplication of integers, fractions and decimals on the number line is more difficult (usually the multiplication here is based on the adding again), and dividing is almost impossible to visualize on number line. Number line seems to support mainly additive thinking (Moss, 2005). The teaching of fractions usually begins with the pictures of pizzas or cakes, divided to the equal pieces, and some pieces are shaded or taken away. The pupils learn to count the number of all pieces and number of marked pieces, and to write these two numbers to the top and to the bottom of the fraction bar as a numerator and a denominator. That kind of approach counting of rational numbers is based on additive thinking as well (Moss, 2005). Further the pupils find out that a particular rational number can take many forms (e.g. 3/5 = 6/10 = 18/30 = = 0.6 = 0.60 = ), and this is a new and odd fact in comparing with whole numbers (Moss, 2005). Duval (2006, p. 108) tells about this same problem in notion changing of the semiotic system:

If for any mathematical object we can use quite different kinds of semiotic representation, how can learners recognize the same represented object trough semiotic representations that are produced within different representation systems?

Almost all of pupils attention goes to the learning these new facts. Calculating with rational numbers is mathematical-technical acrobatics, and it may be guessed that here is not much place to think about measure or relation. Fractions Decimals Percentages Relative (also multiplicative) thinking, fractions and decimals related with percentages, are studied in school mathematics usually in the 6th grade (pupils aged 12 13), and it is a difficult topic for pupils. At first, the additive thinking is deeply rooted in the previous grades. For many years additive thinking has been the norm, and now, in relatively short time (a couple of months) the pupils must apprehend that fractions and decimals are not only numbers on the number line (Moss, 2005) but they embody a relation and a ratio as well (Adjiage & Pluvinage, 2007). Again, the pupils learn fractions and decimals, but at this time in the new sense of ratio, related to the whole. These are another kind of fractions and decimals, although they look the same as earlier. Big amount of rules and algorithms are bound with this topic, and if the teachers teach these rules mechanically, the pupils often do not

know, which of the rules they must use in each case. Talking about fractions, Charalambos & Pitta-Pantazi, (2007, p. 311) say:

instead of rushing to provide students with different algorithms to execute operations of fractions, teachers should place more emphasis on the conceptual understanding of fractions.

The decimals are used too easily as indicator of ratio, and using decimals in this sense in the first stage of learning ratio is too mechanical way of calculating for the pupils. Meaningful use of fractions instead of mechanic using of decimals would be much better (Roche, 2005). The problems with transformations Big amount of exercises on ratio are designed for mechanic training of transformations fraction decimal percentage (e.g. 3/5 = 0.6 = 60%). The skills of transformations are certainly important, because big amount of mistakes are done in this area. It is quite common that when the pupils cannot do transformations correctly, they begin to construct answers with the numbers what they are seeing. Hallett (2008) found that 55% of 13-year-olds answer to the question Which of the following numbers: 1, 2, 19 and 21 are closest to sum 12/13 + 7/8 either 19 or 21. Moss (2005, p. 313) writes:

One of the questions we asked was how the students would express the quantity 1/8 as a decimal. This question proved to be very challenging for many, and although the students ability increased with age and experience, more than half of the sixth and eighth graders we surveyed asserted that as a decimal, 1/8 would be 0.8 (rather than the correct answer, 0.125).

It can be assumed that some types of transformations are simpler for the pupils. In order to understand pupils skills and preferences in transformations of FDP better, I interviewed some pupils (aged 13 14) of 7th grade. A short overview of interview and its results is given in the second part of this paper. Whole System As it is difficult to recognize an object written in different ways, it is also difficult to recognize two different meanings, when they are written in the same way. In both cases we are talking about changing the semiotic system (Duval, 2006) but in different ways. The problem here resembles using homographs in the language. When we have two homographic words, for example party and present, it is impossible to know in which meaning these words are used. Only by adding the third word, birthday or chairman we can firmly say about what the story is about. In the Figure 1 two senses of fractions and decimals are shown: firstly, related with number line they are rational numbers with certain location on it; secondly, related with percentages they express ratio. It depends on context of concrete situation (exercise), in which sense it can use them, and which (additive or multiplicative) is

the strategy of solving. Therefore, in the time when pupils learn rational numbers as expressing the ratio and relation, it is necessary to introduce the WS to them. The key problem is to teach pupils (1) to understand ambiguousness of fractions and decimals (see Charalambos & Pitta-Pantazi, 2007), and (2) to choose right solving strategy (additive or multiplicative).

Figure 1. The whole system.

White, Wilson, Faragher, & Mitchelmore (2007) found in their study that the most complicated lesson (in series of percentage lessons designed by them) mentioned by mathematics teachers was a lesson How do I choose? where pupils compared the appropriateness of additive versus multiplicative strategies. Thus, even teachers feel themselves not confidently when explaining choice of the strategy. It can be guessed that mathematics teachers have not enough good examples and exercises to work on this topic.

THE STUDY In the autumn of 2008 a questionnaire was carried out in seven primary schools in Estonia (with N = 261 children, in 15 different classes) to test seventh grade pupils (age 13-14) skills of percentage calculation. These pupils had learnt percentages in the 6th grade, and in the purpose of the test was to test what they remembered about calculating percentages. Additionally I interviewed 10 pupils from this sample. In this paper I will report results from one of the questions in the interview. This question concerned all 6 sorts of transformations in FDP: Which of these transformations is the simplest? Collate these transformations in order from the simplest to the most difficult. To help pupils answer to this question I used the concrete examples with numbers 3/5, 0.35 and 35%. I presumed that some of pupils would do transformation

Multiplicative

thinking

Percentages

Fractions Decimals

Number line

Additive

thinking

FDL

FDP

3/5 = 0.35 (see Moss, 2005). One boy refused to answer to this question. He claimed that he has forgotten all these transformations. Therefore I have results from nine pupils. All names are pseudonyms.

Table 1. Pupils preferences of performing different transformations. Incorrect trasnsformations are marked with a *.

transformation

pupil

% decimal

35% = 0.35

% fraction

35% = 7/20 or = 35/100

decimal fraction

0.35 = 7/20 or = 35/100

decimal %

0.35 = 35%

fraction decimals

3/5 = 0.6

fraction %

3/5 = 60%

1. Emilia 3.* 1.* 2.* 4. 5. 6.*

2. Harry 1.* 2. 6. 3.* 4. 5.

3. Richard 3. 4.* 1.* 2. 6.* 5*.

4. Harold 1. 6.* 3.* 4. 2. 5.

5. Rebecca 4. 1. 3. 5. 2. 6.

6. Pamela 1. 2.* 4.* 3. 6* 5.*

7. Karen 1. 5. 2. 4. 3. 6.

8. Ken 3. 4. 2. 5. 1. 6.*

9. Norma 4. 3. 6. 2. 5. 1.*

Average 1. (2.33...) 2. (3.11...) 3. (3.22) 4. (3.55...) 5. (3.77) 6. (5)

I analysed the results of answers to this question from two aspects: pupils preferences (Table 1) and skills (Table 2) of transformations. From Table 1 it can be seen in which order the pupils wanted to do these transformations. The most preferred transformation (in average) was from percentage to decimal. The pupils said that it is easy because here must just move the decimal point. Secondly the pupils chose the transformation from percentage to fraction. When the pupil got to answer 35% = 35/100 (without reducing to 7/20), I consider it was correct. The third transformation was from decimal to fraction. Again, the answer 0.35 = 35/100 I consider as correct. The forth transformation was from decimal to percentage. The two latest transformations were from fraction to decimal and from fraction to percentage.

The two basic mistakes in transformations were moving the decimal point only one gap instead of two gaps (35% = 3.5) or a conversion percentage sign not to hundredth but to tenth, (35% = 35/10 = 3.5), and as I was guessed, a combination (3/5 = 0.35 = 35%), (see Moss, 2005; Hallett, 2008). The transformation from fraction to percentage was the most difficult for pupils. Here appeared most of mistakes. (See Table 2)

From Table 1 it seems as if the pupils dont choose the transformations in order of how well they master them. For example four pupils (Emilia, Harry, Richard and Norma) started from transformation in which they gave an incorrect answer. I assume that the question here is not the level of mastery. These pupils had an incorrect subjective knowledge of this topic (Pehkonen & Pietil, 2003), and they were sure that their answer was correct. The transformation from percentage to the decimal had a clear preference in this sample (average 2.33, see Table 1). The next four transformations from percentage to fraction (3.11), from decimal to fraction (3.22), from decimal to percentage (3.56) and from fraction to decimal (3.78) were in medium and quite close-set, while transformation from fraction to percentage (5) seems considerably unpopular.

Table 2. Pupils skills of transformations: right (italic) and wrong (bold) answers.

transformation

pupil %

decimals decimals

% decimals fractions

fractions decimals

% fractions

fractions%

1. Emilia 35% = 3.5 0.35 = 35% 0.35 = 35/10 3/5 = 3 : 5 =

0.6 35% = 35/10

3/5 = 3 : 5 =

0.6%

2. Harry 35% = 3.5 0.35 = 3.5% 0.35 = 35/100 3/5 = 0.6 35% = 35/100 3/5 = 60%

3. Richard 35% = 0.35 0.35 = 35% 0.35 = 3/5 3/5 = 0.35 35% = 3/5 3/5 = 35%

4. Harold 35% = 0.35 0.35 = 35% 0.35 = 3/5 3/5 = 0.6 35% = 3/5 3/5 = 60%

5. Rebecca 35% = 0.35 0.35 = 35% 0.35 = 35/100

= 7/20 3/5 = 0.6

35% = 35/100

= 7/20 3/5 = 60%

6. Pamela 35% = 0.35 0.35 = 35% 0.35 = 35/10 3/5 = 0.35 35% = 3/5 3/5 = 35%

7. Karen 35% = 0.35 0.35 = 35% 0.35 = 35/100 3/5 = 0.6 35% = 35/100 3/5 = 60%

8. Ken 35% = 0.35 0.35 = 35% 0.35 = 35/100 3/5 = 0.6 35% = 35/100 3/5 = 3.5%

9. Norma 35% = 0.35 0.35 = 35% 0.35 = 35/100 3/5 = 0.6 35% = 35/100 3/5 = 0.06%

7/2 8/1 5/4 7/2 5/4 4/5 Average

(right / wrong) 15/3 12/6 9/9

From Table 2 it can be seen that the easier transformations are percentage decimal (15 correct and 3 wrong answers in all), moderately difficult are transformations decimal fraction (12/6), and the most difficult transformations are fraction percentage (9/9). The last of these is quite understandable because transformation fraction percentage needs in fact a mid-transformation fraction decimal percentage, and is therefore more difficult. The strength of the skills in transformations is seen on the Figure 3 in FDP.

THE EXAMPLES FROM REAL LIFE Many authors (e.g. Adjiage & Pluvinage, 2007; De Corte et al., 2005; Moss, 2005) point out a necessity to bring into mathematics learning the examples and exercises from the real life. De Corte et al. (2005, p. 2) write:

Powerful models have at least two important characteristics. First, they are rooted in realistic and imaginable contexts. Second, they are sufficiently flexible to be applied on a more advanced and general level. If models meet those requirements, they can bridge the gap between the informal understanding connected to the real and imagined reality, on the one hand, and the understanding of formal systems, on the other hand.

Figure 3. The real world and the mathematical world.

In the Figure 3 it can be seen that in the mathematical world of the WS there are four axes: number line decimals, decimals percentages, number line fractions, and

Real world

Percentages

Fractions Decimals

Number line

Mathematical world

FDL Rational numbers

FDP Ratio. Proportion

fractions percentages. Below I will investigate briefly each of them separately, and exposit the examples from the real world which would be linked with the WS. Number line decimals On this axis appears the additive side of decimals. Here belong all these examples in which we see various scales: the rulers, the thermometers, the kitchen-scales, the digital scales, the (imaginary) sea level scale, and units of money. On this axis these quantities can above all to add or to subtract between themselves, and when we have two of the same kind quantities, we can them to compare in additive way (subtracting).

Decimals percentages On this axis the decimals take form of ratio. Here belong all the prior examples, when we compare two of the same kind quantities in multiplicative way (part-whole). In addition to them I found some examples from real world where the decimals are ratios in their natural representation. The best example is cash receipts. For example, when there are bought 0.370 kg piece of cheese, on the receipt it is seen as 0.370 times kilogram price of cheese. The second example is money exchange rate. Closer to percentages are increases and falls in price or in salary. Whilst the habitual examples from statistics are usually related with big amounts, then by calculating is necessary to use the calculators, and therefore I guess these are not good examples in the 6th grade (Reinup, 2009).

Number line - fractions On this axis appears the fractions additive side. I sought again examples which were ready in real life, not these where it must the fractions artificially to create. I found just few examples from real life. The best example was a calendar with 12 months in a year, with 30 days in a month, and with 7 days in a week. Second example was a clock with 24 hours in a day, with a quarter after or past, and half past a full hour. A pizza or a chocolate bar with equal pieces belong here as well, although in my opinion it is a little artificially created situation. Fractions percentages Here the fractions have got the classical part-whole sense, the sense of quotient. Very many of examples found here: the concentration, the zooming and minimizing, increase and fall in price (in cases using the fractions), and all sorts of probabilities.

THE CONCLUSIONS AND PRINCIPLES FOR DESIGNING OF EXERCISES From the theory it is known that the calculating with the fractions and decimals is difficult to the pupils (Moss, 2005; De Corte, Depaepe, Op t Eynde & Verschaffel 2005; Adjiage & Pluvinage, 2007). The difficulties are caused on one hand (1) from ambiguousness of fractions and decimals (Charalambos & Pitta-Pantazi, 2007) and

necessity to change from one semiotic system to other (Duval, 2006), and (2) from the low level of the skills in transformations of FDP (Hallett, 2008; Moss 2005). The pupils interviewed by me prefer transformation in succession first from the percentage to the decimal and fraction, second from the decimal to the fraction and percentage, and last from the fraction to the decimal and percentage (Table 1). Their skills of transformations were the best in percentage decimal, medium in decimal fraction, and the worst in fraction percentage (Table 2). The typical mistakes were the mechanical moving of the decimal point, and combination of numbers. Alarming in these results is the fact that in transformations fraction percentage where the fractions are used in their classical part-whole sense, the pupils skills of transformations are the worst. The adults divide the problems of ratio into many different subtypes (Charalambos & Pitta-Pantazi, 2007; Adjiage & Pluvinage, 2007). In my opinion instead this in the 6th grade, when the pupils are first time learning the ratio, it is important to show only two ways in principle: the additive and the multiplicative way. As a Big Idea, Brigham, Wilson, Jones & Moisio (1996) suggest in teaching of fractions, decimals and percentages to use ratio or, in simpler terms, division as well. Knowing the theoretical base, the pupils skills in transformations, and the examples from real life, linked with the WS (Figure 3) one can design suitable exercises for teaching of the WS. These exercises:

are desirably exciting or humorous situations (from real life); afford to ask many questions related to this situation, which include both

additive and multiplicative ways of thinking. Exciting or humorous situation calls forth interest to find the solutions (Schweinle, Meyer, & Turner, 2006). Asking consecutively questions, involve both ways of thinking, shows to the pupils better the ambiguousness of fractions and decimals, which is actually quite normal in real life. For expressing of ratio it can suggest using fractions rather than decimals, because it improves the pupils skills of transformations on axis fractions percentages (see Table 2; Figure 3). Next, concrete exercises for the teaching of the WS must be designed. Through an empirical study we will investigate, if these kind exercises have an effect in the teaching of WS, and are they helping pupils to understand better the choice of right solving-strategy.

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