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The report of a small study on percentages

5 October 2012


  1. Introduction

This paper is the report of a small study on percentages conducted in a seventh grade of one international secondary school in Rotterdam from September until October 2011. The study was conducted as a part of one course the researcher took, namely Integrative Practical (IP). The course aims at giving experience to master students in doing a design research. The study aims at supporting seventh graders in developing their understanding on percentages. In the light of it, the general research question posed in this study is how can we support pupils in developing their understanding of percentages? This general research question can be elaborated into three specific sub questions:

  1. How do seventh grade pupils solve percentage problems?
  2. How can models support pupils’ understanding of percentages?
  3. How can pupils learn to understand using fractions and percentages as means for proportional comparison?

As additional information, in this study, the researcher would only focus on one pupil, namely Nouran.

  1. Data collection

The study was conducted in The American International School of Rotterdam (AISR) from September until October 2011. The school, AISR, is located on Verhulstlaan 21, Rotterdam. It has multicultural students from pre-school to grade 12 and uses English as main language in teaching-learning activities. This study was a part of a larger study whose subject was comprised of 14 seventh graders who already have experience with percentages. Related to the larger study, these 14 pupils did written test about percentages, then they were interviewed about their solutions, before finally got 2 x 45 minutes lessons about percentages in groups of 2 or 3. As already mentioned in the previous section, in this study, the researcher would focus on one pupil, Nouran. In the two percentages lessons, she was in a group of 3 with two other pupils namely Andy and Jiwon.

In this study, the researcher, together with one other researcher who was doing the same small study but with different focused pupil, only worked with a group of three pupils consisted of Andy, Jiwon, and Nouran. The researcher collected pupils’ answers for written test on percentages, video recordings and notes of their interviews, and video recordings and notes of two 45-minute lessons they got. The first data, pupils’ works on percentages written test, were collected on September 2011. The pupils were given written test on percentages consisting of 15 items arranged from the easiest to the hardest one. This test was given in order to get information about pupils’ prior knowledge related to the percentages. The second data, video recordings of pupils’ interview, were collected one week after the written test. The interview was conducted by the researcher and his colleague in one small classroom. They interviewed 2 or 3 pupils. Every pupil was interviewed one by one for about 15 minutes by one researcher while the other one took a note and made sure the video camera used to record the interview worked well. This job included pushing the record/stop button in the camera. The interview was aimed at getting elaborations of pupils’ solutions for the written test in that how and why they used certain procedures to solve percentages problems. The interview would also give more information about what the pupils already understand and what they do not understand yet. In the 15 minutes interview, every pupil was asked to do problems number 6 (Figure 1), percentage of sports member, and problem number 9 (Figure 2), reduced-price bike, taken from the previous percentage test. The pupils should explain their understanding about each problem and the way they solved it. In doing so, they were also given written form of these two problems. Each problem was given in different paper, so the pupils could write down their thought about each problem while being interviewed.



Figure 1. Problem 6             Figure 2. Problem 9

The last data, video recordings and notes of two 45-minute percentages lessons, were collected on 2 different days. The second lesson was given one week after the first one. In these two lessons, the researcher also worked together with the same researcher he worked with in the interview. In each 45-minute meeting, three pupils were given lesson by one researcher while the other researcher took a note and made sure the video camera used to record the lesson worked well. The lessons were designed based on the result of pupils’ written test and interview. The goal of the lessons in general is to support pupils in developing their understanding of percentages. The video recordings were needed to analyze whether the lessons already fulfilled the goal and also to analyze pupils’ interaction in the lessons. For the first lesson, two problems were given to the pupils, namely Hogwarts Mathematics Club (Figure 3) and Sales Tax (Figure 4), while for the second one, there was only problem given, Aiming for the basket (Figure 5).

Hogwarts secondary school has 500 students. 35% of those students are members of mathematics club. How many students in that school who are the members of the mathematics club?

Mr. Smith would like to buy a new laptop. For that laptop, he must pay 7% sales tax.

How much Mr. Smith must pay for the tax?

Figure 3. Hogwarts Mathematics Club

Figure 4. Sales Tax Problem


Figure 5. Aiming for the basket problem

  1. Lesson Design
  2. Hypothetical learning trajectory (HLT) lesson 1

    HLT Lesson I

    Learning goals

    Pupils are able to use model of percentage bar as a model for thinking and reasoning in solving percentages problems.

    Starting points

    • The pupils can represent the part whole relationship using fraction
    • The pupils could solve some percentage problems using mental calculation
    • The pupils have some strategies in solving the problems, such as using fraction, decimal, and proportion
    • The pupils have difficulty in doing percentage problems that have “difficult” number which is defined as a number not divisible by 100
    • The pupils do not know how to use the model of percentage such as percentage bar or table, which can be used as a tool to reason with
    No

    Instructional activities

    Assumptions about how these activities support mental activities that lead to the overall learning goal.

    1.

    Problem 1:

    Hogwarts Mathematics Club


    Hogwarts secondary school has 500 students. 35% of those students are members of mathematics club. How many students in that school who are the members of mathematics club?

    • The pupils will try to find the 35% using their own ways such as using the formal one 35/100 x 500
    • The pupils can use fraction, decimal, or proportion to solve the problem
    • The 500 as the total make the pupils easy to use proportion 35/100=…/500
    • The 500 as the total make the pupils easy to work with benchmark numbers such as 50%, 25%, and 10%
    • The pupils will relate 35% to benchmark numbers they already familiar with: 25% and 10%
    • The pupils will use percentage bar to show the positions of 25%, 10%, and 35% and make relation amongst them
    • The pupils will compare solution they get by using their own ways with solution they get by using percentage bar and choose which one is easier.

      The pupils will use percentage bar to find a certain percents of a whole by first finding related benchmark numbers

    2.

    Prob Problem 2: Sales Tax

    Mr. Smith would like to buy a new laptop. For that laptop, he must pay 7% sales tax.

    How much Mr. Smith must pay for the tax?

    • The pupils will try to find the 7% using their own ways such as using the formal one, 7/100×750
    • 750 as the normal price and 7% as the tax will make the pupils difficult to do it in the formal way
    • The proportion way is also difficult to be used in this situation since 750 divided by 100 does not yield an integer
    • The pupils will use percentage bar to solve the problem since they have already learnt it when doing the first problem. They will find difficulty when only referring to the benchmark numbers such as 50%, 25%, or 10%
    • With the help of percentage bar, the pupils will find that by finding 1% of the 750 they can easily find the 7% and other percents.
    • The pupils will recognize 1% procedure as a strategy that always works in every situation, but not always as the easiest one

Based on the written test and interview session with the three pupils, they already have their own strategies to work with percentages problems, so they will use their own strategies to solve the given problems. This is also the researcher’s intention to allow the pupils solving the problems by themselves. After working with the first problem, the researcher who is also the teacher in the lesson will ask the pupils, one by one, to explain to the others how he or she can get the answer. It is conjectured that the strategies of each pupil will be different. After that, the teacher can introduce the use of a bar model in solving problems. Teacher brings pupils to find the benchmark percents that can be presented in the bar. The usefulness of the bar will be developed to make the pupils aware to the basic operations (e.g. addition, subtraction, and multiplication) can be done using the bar. These operations later on will help the pupils to find the number of percents that is asked. For example, the pupils can find 35% by first finding 25% and 10% before adding the results to find the final answer.
The pupils will use percentage bar to show the positions of 25%, 10%, and 35% and make relation amongst them.    The teacher will ask the pupils to compare solution they get by using their own ways with solution they get by using percentage bar and choose which one is easier.

Using the help of the bar model, the teacher also can promote the idea of 1% procedure in solving percentage problem to the pupils. The teacher will give some questions which are the elaboration of the first problem. The pupils are asked to find 5%, 7%, and 3% using the help of the bar model. The teacher then asks pupils opinion about the name of the bar. It is predicted they will name the bar a percentage bar. This activity will lead the pupils to solve the second problem by working with 1% procedure using the percentage bar. If the pupils come with the fraction or decimal strategy for the second problem, the teacher will ask them to try using the percentage bar to make sure they can use the model.

  1. Hypothetical learning trajectory (HLT) lesson 2
HLT Lesson II

Learning goals

  • The pupils are able to solve a relative comparison problem using fractions or percentage
  • The pupils understand fractions and percents as a means for proportional comparison
  • The pupils are able to change fractions into percents and vice versa

Starting points

  • The pupils can represent the part whole relationship using fraction
  • The pupils can solve some problems by using mental calculation
  • The pupils have some strategies in solving the problems, such as fraction, decimal, and proportion
  • The pupils understand and can use well benchmark numbers such as ½, ¼, 50%, 25%, and 10% along with their relations
  • The pupils can use 1% procedure
  • The pupils know how to use the model of percentage such as percentage bar or table, which can be used as a tool to reason with.
  • The pupils have difficulty in doing percentage problems that have “difficult” numbers which she defines as a number not divisible by 100

Activity


No

Instructional activities

Assumptions about how these activities support mental activities that lead to the overall learning goal.

1.

2.

Finding child who has scored best

Ordering the children based on their shootings

  • The pupils will think that the best is the one with most black dots
  • The pupils will consider the role of white dots
  • The pupils will make fractions for the number of black dots/total
  • The pupils will make fractions with the same denominator (200, or even 100)
  • The pupils will use the half as means for making a decision
  • The pupils will make percentages of all scores

The teacher will allow the pupils to work with this problem for several minutes before leading the group discussion. The pupils will be asked to explain their strategy in solving this problem. If there is a pupil who thinks that he or she only needs to consider the number of black dots to determine the child scoring best, teacher will ask the other’s opinions about this. If all pupils do the same, only considering the black dots, teacher will ask them to pay attention to the white dots and its role in this problem. In determining the child having scored best, there is a possibility the pupils will use the idea of half as the base for comparison. The teacher will ask the reason of doing that before leading to the discussion about the use of percentages to check the correctness of their answer.

If the pupils have difficulty in solving this problem, the teacher could suggest them to use the model that has been learnt in the previous lesson, percentages bar. If the pupils solve the problem using fraction with the same denominator and do not have difficulty in calculating, the teacher then promotes the use of percentages to check whether the solution will be the same or not. At the end of the lesson, the teacher and the pupils will draw a conclusion about the most efficient way to do a comparison.

  1. Data analysis
  • Written test

Here will be given the analysis of Nouran’s work for the written test. The analysis was done by taking into account all that have been written by Nouran in the test including solutions and procedures carried out to come to the solution. She could correctly answer the first 5 questions of the written test and also question number 12, so at the total she made 6 correct answers out of 15 items tested. The first five questions are the easiest ones since the problems are arranged from the easiest to the hardest one. Looking at what Nouran did for the first five questions, it can be concluded that she already knows that 100% means a whole and one whole is 100%. There is evidence from her solution for question number one asking about how many percents is the cotton of one pair of socks if it is known that the socks were made only by 43% of polyamide and certain amount of cotton stated in %. She wrote down 100% – 43% = 57% to get the percentage of the cotton. What she wrote down in question number 2, 3, 4, and 6, asking about percentage of a part related to a certain whole, shows that she understands the part whole relationship in percentages and also the meaning of percentages. Her solution for the question number 5 shows her ability in converting percentage into fraction. She wrote 0.5%=5/1000=1/200. She got the 200 by dividing 1000 by 5. In answering this question, she ignored the given pattern. Even it is written in the problem that 50%=1/2 and 5%=1/20, she used her own way to get the fraction of 0.5%. Considering the prior knowledge she already has, it was surprising that she could only make 6 correct answers and did not give any answer for 4 problems. One explanation that could be derived from her work in the written test is she could not do the calculation if the numbers included are not easily computed. For example, in one problem she could correctly give 30/100×100=30% but failed to give a final answer after writing down 15/20×100= in another problem. Her problem in calculation also appears in her work for problem 9, asking the new price of 600 euro bike after 15% reduction. Although she correctly wrote down that the reduction equals to 15/100 x 600, she gave 190 as the result of the calculation. Another problem that Nouran has is about the procedure to convert fraction into decimal. There is evidence found in number 6 when she tried to convert 15/20 into fraction. Instead of dividing 15 by 20, she divided 20 by 15 to give 1.33 as the result of the calculation, but she did not give this number as the answer for number 6 and just left it 15/20×100=. Probably her percentages sense doubted 1.33 as the result of 15/20. The fact that she could give the right answer for problem 12, asking 35% of 2800, by writing down 35/100×2800=980 gives more evident that she has more problem with the calculation not with the percentages it self, so she needs a model that could support her in doing the calculation and the bar model of percentages is an appropriate one.

From the analysis of Nouran’s written test, it can be seen that she has a good basic knowledge about percentages. She already knows that 100% means a whole and one whole is 100%. She also already understands the part whole relationship in percentages and also the meaning of percentages. Related to the percentage skill, she shows her ability in converting percentage into fraction. Nouran’s problem is about the calculation. She could not do the calculation if the numbers included are not easily computed. In addition she also has problem in converting fraction into decimal. It can be concluded that she has more problems with the calculation not with the percentages it self, so she needs a model that could support her in doing the calculation when dealing with percentages problems and the bar model of percentages is an appropriate one.

  • Interview

Here will be given the analyses of Nouran interview for problem number 6 (Figure 6) and 9 (Figure 7). The analyses were done by taking into account the video recording along with its transcription and the note taken when the interview was conducted.



Figure 6. Problem 6             Figure 7. Problem 9

Problem 6

The interview was started by giving problem 6 on a paper to Nouran. After reading the problem, she started by writing down 15/20×100=. She explained that first she found the total number of the students which is 20. To find the percentage, she put 15, which is the number of students being members of the club sports, above 20 to form a fraction and multiplied it by 100 to get the percentage. Asked why she multiplied by 100 to get the percentage, she reasoned that % means 100. When she did not know how to convince the interviewer, she just said that she learnt the procedure to make fraction into percentage, putting the number over the total then times 100.

Even though she knew well the procedure, she could not come into the right answer. She has a problem with calculation and she herself admitted it. She used to do calculation with calculator. One reason that she gave, the final number is not important, since she could find it using calculator. She said what she thinks important, “the whole is the base then we have to make the process”.

She tried to do long division to get the decimal of 15/20, but she made a mistake. Instead of dividing 15 by 20, she divided 20 by 15 and she also gave wrong result of the division, 0.121212. Then she multiplied it by 100 to give 12% as the final answer. Asked whether she is sure or not with the answer, she said not sure and immediately judged that her answer is wrong. The reason is very interesting, she said “Because actually half of twenty is ten. If it’s a half so it’s 50% and right now, it’s more than half. So it cannot be like this, I know”. This reasoning shows her understanding about percentage, half, and 50% as benchmark number. She just does not know how to calculate 15/20 by hand.

Given an easier fraction, ½, by the interviewer, she without any calculation knew it equals 0.5, but when the interviewer asked her to perform the calculation in long division, which was intended to lead her that 15/20 means 15 divided by 20 not the reverse, again she failed. She wrote down both, 1 divided 2 and 2 divided by 1, but could not do correctly any of them.

Realized that she had good sense about percentage, but also had problem with the numbers at the same time, the interviewer tried to change the question in the problem. The interviewer asked about percentage students who do not choose sports since the fraction will be easier, 5/20. This question worked well, Nouran immediately recognized that it equals ¼ which is 25%. She also concluded that the solution of the initial problem is 75% that she got from 100% – 25%. Get surprised by the easiness of the problem, she laughed and hit the table. She also mentioned that 15 is 75% of 20 and this time she was really sure with her answers. At the end of the problem 6, she explained that 100 which she subtracted by 25 is not the number of students. Instead, it is like the whole, the total of students in a portion.

Problem 9

In doing problem 9, Nouran began with writing down 15/100×600. She referred it as what she had to pay after the 15% reduction, although she just explained that 15% is the discount. After doing some calculation, she got 90 as the result which also she gave as the solution of the problem, but she herself surely said that 90 is a wrong answer. Again as she did for number 6, she used 50% and half for her reasoning. She said “Because, ya.. because.. if it’s like 50% reduction, it would be 300 euros that we’ll pay..” and “Ya, cause 50% means the half of 600. So, it’s gonna be 300 euros if it is the half and actually reduction is less than 50%, so we have to pay like more than 300..”. The interviewer asked her again about what 15% reduction means for sure, and she said that it means taking off 15% of the normal price.

Nouran tried to utilize another approach to solve the problem. She made such a price-reduction table, but she could not finish it. She said ever worked with it last year, but forgot how to use it to solve the problem. To help her, the interviewer modified the question by changing the reduction to an easier number, 10%. Again, it worked well. By considering the 10% reduction that she could easily count mentally, she realized that 90 that she got before is not the price that she should pay. Instead, it is the reduction, amount that she should take off from the normal price. So, she took 90 from 600 to give 510 as the solution of problem 9. She also gave 540 as the answer for the new problem that she got from 600 minus 60, ten percents of reduction that she counted mentally. For the last time, the interviewer asked her what is the price if there is 100%, and she with full of confident said zero.

Relation between Nouran solution for number 6 and number 9

What is similar in Nouran’s ways to solve problem 6 and 9, she always used half, 50% to verify her answers. For number 6, she rejected 12% that she got after considering that half of 20 is 10 and half is 50%, and realizing that the members of the sport club is 15 which is more than 50%. For number 9, she judged that 90 is the wrong answer by arguing if the reduction is 50% which equals half of 600, the price would be 300 euro. Since the reduction is less than 50%, she thought she has to pay more than 300 euro.

The next relation between her solutions is she always could come to the right answer if the numbers are familiar or easy to compute with her mental calculation. For number 6, she came to the right answer after the question had been slightly changed. It made the fraction representing the problem changed from 15/20 to 5/20 which she easily recognized as ¼ and 25%. For problem 9, she gave the right solution by recognizing what 90, which she got from 15/100×600, means after the reduction had been changed from 15% to 10%.

Here is a part of transcription illustrating Nouran’s solution procedure for problem number 6. It is taken as an example how Nouran solves problem related to percentages. She mostly uses a very formal way, by forming a fraction of part and whole and multiplying it by 100. From this part of transcript, it also can be seen Nouran’s problem in converting fraction into decimal.

(I stands for the interviewer and N for Nouran)

N Ok, what percentages of the children in class five is a member of a sports club? There is a question about class 5, people who are member is 15 and not member is 5
I Ok, how do you do this problem? This is a pen
N (silent) if there are all twenty, cause we have 15 and 5 so there all are 20, and people who are member of sports club is 15, so it is 15 over 20 times hundred to get the percentages
I Why you should multiply it by 100 to get the percentages?
N Cause actually this means 100 (pointing to the percent sign), it’s like the percentages mean 100.. its like…
I just learnt it when I want to make it to percentages I put the number over the total times 100
I Hmm.. so you will get the percent
N What?
I So, after you put the number one above another and then times by 100, you will get the answer?
N Ya..
I So, would you mind finishing this problem?
N Ok, Actually I am not really good doing this
I Ya, it’s Ok
N …just make it with calculator, it’s easier.. (silent)
I So, you are used to working with calculator?
N Actually, I’m new here this year, but I used to use that in my old school
I Ok.. It’s Ok, you can try to do it. We have a lot of time..
N (silent)
I am not gonna make a correct answer.
I Yah, it’s Ok.. we do not want to find the correct answer, we just want to know how do you do this kind of problem. So, what are you doing now?
N I’m trying to divide them (dividing 20 by 15)
I Hm.. dividing..
N (silent while writing on the paper) It’s gonna be infinite number.. (she got 0.121212)
I Ow, infinite number? So, you divide 20 by 15 ya?
N Ya..
I And why do you do like that?
N Because actually it is here a fraction and that line is division, so be like divided and the result which come to us will multiply by 100 is actually like is infinitive right now… So, if I make it, it’s gonna be 0.121212… so let’s just multiply it by 100, it’s gonna be like 12… (writing the paper 12.1212 ~ 12%)

From the analyses of the interview, a diagnosis about what Nouran already knows and what she does not know can be made. Here is the list.

What she already knows and understands?

  • She understand the percentage
  • She understands that the whole is 100%
  • She knows the procedure to compute percentage of a certain part related to a whole
  • She knows the procedure to compute a certain percent of a whole
  • She could use complement to solve percentage problem,
  • She knows reduction means discount, taking off price, or sale
  • She understand and could use well benchmark numbers such as half, ½, ¼, 50%, 25%, 10% along with their relations
  • She is more interested in process than the result.

    What she does not know and understand yet?

  • She does not really understand that fraction for example 1/2 means 1 divided by 2
  • She could not solve percentage problems that have ‘difficult’ numbers, which she defines as numbers that she could not compute mentally
  • She does not know how to use model of percentage such as percentage bar or table, which can simplify the calculation

    According to this diagnosis, two lessons can be designed. The goal of the first lesson is the pupils are able to use model of percentage bar as a model for thinking and reasoning in solving percentages problems and the goals of the second lesson are (1) pupils are able to solve a relative comparison problem using fractions or percentage, (2) pupils understand fractions and percentage as a means for proportional comparison, and (3) pupils are able to change fractions into percents and vice versa.

    Hogwarts secondary school has 500 students. 35% of those students are members of mathematics club. How many students in that school who are the members of the mathematics club?

    Mr. Smith would like to buy a new laptop. For that laptop, he must pay 7% sales tax.

    How much Mr. Smith must pay for the tax?

    Figure 8. Hogwarts Mathematics Club

    Figure 9. Sales Tax Problem

    • Lesson 1

    As already mentioned in the data collection section, the focus of the analyses on the 2 lessons given is only on the learning of one pupil, Nouran. The first lesson is aimed at the pupils are able to use model of percentage bar as a model for thinking and reasoning in solving percentages problems. The lesson was started by giving Hogwarts mathematics problem to the pupils (Figure 8) and followed by giving sales tax problem (Figure 9). Here will be given analysis of Nouran’s learning activities in the lesson 1. Just after finishing Hogwarts mathematics problem, Nouran gave an explanation about what she knew from the problem. She said “If there are 500 students in the school, 35% of them are the member of mathematics club, so 35% means 35 over 100”. In addition she also proposed her way to solve the problem by saying “So, to get like the number of member 35 over 100 which is 35% and multiply with the total number which is 500 and give the result 175 students”. Here can we see the typical of Nouran’s solutions which matches with what is already conjectured beforehand in the HLT for the first lesson. She always solves percentage problem using a very formal way, just like what she did in the written test and interview. She does not consider any model of percentages to solve the problem. It is because, according to the information given by her in the interview, that formal way is exactly what she learnt in the previous grade. She also added that her teacher at that time did not directly teach the formula to calculate the percentage, but she must find it by herself while working a structured worksheet about percentage. Since she could not always come to the right solution while using that formal way, especially if the numbers involved are not easily computed, the idea of percentage bar would be introduced to Nouran.

    Nouran gave her participation at the first while together with 2 other pupils, Andy and Jiwon, constructing their first percentage bar. She said that 100% of the students equal to 500 students and wrote it down at the right end of the bar that was already drawn on the blackboard. She also added some information by saying “This is 500 students for 100% so if we want to shade the bar. It’s gonna be the whole”. Asked to find another percent of 500 using the help of the bar and given example 50%, Nouran gave her fast respond by saying 250. It was not surprising since in the written test an interview she always connected her work to some benchmark numbers especially 50% and a half. Asked to make it in the bar, she went to blackboard and drew the line for 50% in the middle of the bar while saying “Ok… you can divide it, it’s like you can use ruler to make it equal. So, this is 50% and 250 students”. She also gave explanation about how to come to 250, she said “500 divided by 2” and “Like if we have 100 and we want to have a half of it, we divided by 2 and we get the result”. In addition she also gave the way to get 10% and 25% of 500 which is dividing 500 by 10 and 4. It seems that she does not need the help of the bar to get the 50% , 10%, and 25% of the 500. She used it since the teacher who was also the researcher asked her to do so.

    Nouran really made use the bar when trying to give another way to get 75% of 500. She said “Um… because if you divide 500 by 4 you get 125, and like we have 50%.. that’s 250, so we add 125 and 250 that will be 375”. She never used this kind of approach before. Another nice thing is she was the first pupil naming the bar as percentage bar when they were asked about what they would call the bar. At the end of the first session, she also visualized 5% of 500 with the help of the bar.

    In the following session, before working the sales tax problem, the pupils were asked to find 3% of 500. In this time, Nouran with her formal way failed to get into the right answer. After writing down her solution on the given paper (attached), she said “If like the whole, the school has 500 students and 3 over 500 and multiplied them by 100 to get the percentage. We divide 3 by 500 and the result we multiplied by 100 and actually the result is 0.161616… and then we multipled by 100 it is gonna be 16.1616..”. This kind of mistake ever occurred in the written test and interview. It is caused by the numbers that are not easily computed by Nouran. After given help to consider the bar and asked to find 1% first, Nouran could answer correctly that 3% of 500 is 15. She got it by multiplying 5 which is 1% of 500 by 3. She also could give 7% and 13% of 500 by the same reasoning.

    Given sales tax problem asking 7% of 750, Nouran could give the right result without the help of the bar. She again used her formal way which is also already predicted in the HLT for this problem. She explained “I like… um.. if 100% equal 750 Euros. So, 7% equals how much and then I made a kind of scissors… 7 times 750 divided by 100 and then I get the result 52.5”. Even she did not use the bar, she showed the idea going to 1% just discussed. She also added “It’s like… it’s easier using mathematical operation…”. Given one last question about sales tax problem, 75% of 750, she kept doing it by finding 1% of 750 then multiplying it by 75. Asked whether she will always use that way, she said “Well, it depends.. It’s like if one way doesn’t work, I will try another, maybe like to get 50% first or something will be easy then I will use the way or something that can easily get from 100%, or maybe to get 1% first will be easier, then I will do that way..”. Her explanation shows that she preferred to work with benchmark numbers, what she showed from the beginning, before deciding to use going to 1% that always works. At the end of the lesson, she gave her impression about the percentage bar. She admitted that it is easy to use the bar but she already has another way that she already gets used to. She said “It’s like. I.. It’s like.. we always do with it like writing numbers and so on, that’s what I got used to, so maybe like cause this is a new way like.. or something that I have got used to it. But it’s kind of easy, like it’s easy…”. From this lesson it can be concluded that Nouran could use model of percentage bar as a model for thinking and reasoning in solving percentages problems but does not prefer it for doing so. The reason is she already gets used to work with the formal way, even it usually leads her to the wrong solution.

    • Lesson 2

    The second lesson has three goals: (1) pupils are able to solve a relative comparison problem using fractions or percentage, (2) pupils understand fractions and percentage as a means for proportional comparison, and (3) pupils are able to change fractions into percents and vice versa. It was started by giving aiming for the basket problem (Figure 10) asking the pupils to determine the child having scored best and to order all the children based on their scores.


    Figure 10. Aiming for the basket problem

    Here will be given analysis of Nouran’s learning activities during the second lesson. Given time to read and think the problem for herself, Nouran directly counted the number of black-dots of each child and wrote it on the right end of each child’s dots. She also wrote down Bram, who has the most black-dots among the others, as the child who scores best. This kind of solution is already predicted in the HLT for this lesson. It can be interpreted that Nouran saw the problem as an absolute comparison, not as a relative one as expected. In addition, she also ordered all children from one having the fewest black-dots to one with the most black-dots. Even Julia and Paul have the same number of black-dots, 10, she put Julia before Paul in her order. It is not because she considered also the total dots children have. She did it since Julia comes before Paul on the order written in the problem.

    After having enough time looking at the problem, Nouran and two other pupils were asked about what they know about the problem. Nouran took the first chance. She explained what she knew and also her solutions for 2 questions in the problem. She said “It tells us about basketball play, children are scored like… dots.. black dots it means they scored.. a… it’s like the child who had scored the best was Bram and um..if we want to put in order we can say Liz, Kim, Tom, Julia, Paul, Tess, Ernst and Bram”, exactly like what she already wrote down in her worksheet. Two other pupils also gave their responds about the problem and both of them gave relative comparison idea. Jiwon talked about how many scores each child makes out of a certain number of attempts while Andy also considered the total attempts each child made. Asked to focus only to the first question, asking who has scored best, Nouran kept her belief about the absolute comparison, even she already heard Jiwon’s idea to make fractions of each child, which its numerator is the number of black-dots and the denominator is the total dots, before making all the denominators same and finding the largest. Nouran said “I have ..a… like another idea.. to take care of them.. well, actually it doesn’t matter times he tried to score but what matters and what it asked in the question is how many times did he score. So, if we need like the most person we can count like each one how many times that he score and then we gonna got the most score.. So, I don’t think like matter how many times he tried to score”. Although another pupil, Andy, also tried to convince her to take into account the total number of dots by talking about the bar, using half of the total number of shoots for reasoning and reversing the question by asking how many times they did not score, Nouran did not change her mind. She only said “I know what’s like .. I don’t feel like considering the denominator like.. that the way task. I don’t feel it is important.. cause we need just like the number times they scored but we don’t need the number they have try”. She understood what Andy tried to explain to her and admitted it was right. Nouran’s respond to Andy’s explanation showed that she understands the idea of relative comparison, but did not use this idea to solve the problem since she did not feel the question asking about that. Even Jiwon also tried to help Nouran by saying that she is right if all children have the same number of attempts, Nouran kept her belief about the problem until the teacher, in the end of the first session, “pushed” her to accept that she needs to consider the total number of attempts in order to make a fair comparison.

    In the following session, all pupils were asked to try answering the problem using what they learnt in the previous week, the percentage bar. Given the task to find Ernst’s score percentage (20 out of 50) using percentage bar, Nouran could draw the bar of Ernst and gave 40% as the answer (Figure 11). She started by drawing 50% line which equals to 25. As often mentioned before, she could work well with benchmark number such as 50% or half. For the next step, she correctly drew a line for 20, in the left of 25-50% line (Figure 11), but in finding the percentage of 20 she did not use the idea of the bar. Instead, she used her formal way by writing down 20/50 x 100 = 40%. She also could find the percentage of the second child, Julia, (10 out of 20) with the bar, but she did not really need it since the number is very easy. By the end of this session, after Andy explaining his idea about only using the half of the bar to determine the best scoring child, Nouran agreed that Tess is the child who has scored best and she said “Ya, I agree but it’s like.. what Andy did.. like if you wanna compare but it’s not like by comparing who has before the half, after the half, but.. actually compare..he should first compare by making the denominator the same then he can do it.. you know? Like if you wanna said maybe you can do it on 100…”. She knew that in order to find the child scoring best she could make all children’s scores-attempts into fractions and find the same denominators, that she proposed 100 which equals to the percentage idea, before comparing all of them.


    Figure 11. Nouran’s Percentage Bars

    Nouran tried to apply her idea about finding the fraction of each child to answer the second question, about the order of the children. She could make the correct fractions, except for Julia that she overlooked as 10 out of 40, but she found difficulty when comparing all the fractions. It is strange that she did not try to find the same denominator since she just said that idea to the others. When the teacher suggested her to find the percentage of each fraction, Nouran tried to use her formal way, for example by calculating 9/25 x 100. Using that formal way, she could only correctly find percentages of 4 fractions (out of 8 fractions involved). Just like what she did in the beginning of this research, she knew the procedure, part divided by whole times 100, but often failed in finding the final number. She also did not want to use the help of percentage bar since did not find it is easier. Asked whether she knows that percentage from a fraction can be obtained by making its denominator into 100, she said that she does not know that. She tried to make the denominators same but finding difficulty since the numbers were not easy in her opinion. Suggested to make all denominators into 200 as Jiwon had explained in the whiteboard, Nouran could do it correctly for all fractions as Jiwon did, but she did not order them.

    In the last session, all the pupils were encouraged to use percentage to order the children. Based on Jiwon’s work on the white board, making all fractions into the same denominators, 200, they together were expected to find the percentage of each child. Trying to get the percentage of the first child, Ernst, having fraction 80/200, Nouran could not answer it directly, she did not see that she could divide both the numerator and denominator of the fraction by 2 to make it into hundredth, which equals to percents. Instead, she kept writing 80/200×100 in her paper and later on found 40%. Getting another turn to find Bram’s which was written as 200/400 on the white board, she could find 50%, Asked about the reason, she said “because 10 over 20 is half”, even Jiwon just said that Ernst’s is 40% by dividing both the numerator and denominator of the fraction by 2. Again Nouran showed her ability to work with a half and also her tendency to keep on her belief. In the next turn when finding Bram’s, even after Andy explained Tom’s using dividing by 2 reasoning as Jiwon did, Nouran still could not find it easily. She spent much time and kept writing down 96/200×100 before finally giving 48% as the answers. At the end of this session, after getting all children’s percentages and ordering them together, all pupils were asked about which strategy is easier to use in order to compare all children’s scores and order them and here is Nouran’s respond. She said “Ow, actually it depends maybe be there are some numbers we don’t have the common denominator, you have to make it into percentage then…maybe like some numbers are hard to get their percentages, it is easier to get the common denominator, you know? Something that you get from fraction. It depends on the numbers”. Here can be seen that Nouran knows the way to solve the problem. She understands the idea of relative comparison but usually fails to get into the solution. One explanation about this finding is she was taught that the most important thing is the idea behind the problem and the last number which is the solution of the problem is not important, as she explained in the interview section.

    From the analysis of the second lesson, it can be concluded that Nouran is always able to find her own way, usually the formal one, to deal with a percentage problem, but it is not always easy for her to do it in her way, especially if the numbers involved are not easy. She scarcely wants to use strategies suggested by others. Related to the learning goals of the second lesson, even Nouran did not see the best scoring child problem as a relative comparison problem and did not solve it using fractions or percentage, some her explanations showed that she understands the idea of relative comparison and also understands that fractions and percentage can be used to solve proportional comparison problem. In addition, she is also able to change fractions into percents.

  1. General conclusion and discussion

After analyzing all data in the previous section, answers for the research question and its three sub questions will be given in this section. Nouran, a seventh grade pupil, tends to use a very formal way to solve percentage problems. She prefers to use the procedure “part divided by whole times 100”, that she already learnt in the previous grades, to get the percentage of a part related to a certain whole. For example in finding percentage of sports club members if there is known that 5 out of 20 students are the members of the club, she immediately starts writing “5/20 x 100”. The use of this procedure is not wrong but it usually leads her to the wrong solution, especially if numbers involved in the problem are not easily computed mentally. Other strategy Nouran uses is working with benchmark numbers such as 50%, a half, ½, 25%, and 10%. She uses this way since already familiar with these numbers in her daily lives. She clearly knows what 50% or 25% discounts means. Another reason, these numbers are easily computed mentally by Nouran. She does not need time to say that 50% of 600 is 300.

The use of percentage bar model can support seventh grade pupils’ understanding of percentages in that it can simplify and visualize the calculations needed and can serve as a model to think with. Even Nouran does not prefer to use this model, in the first lesson, she showed that the percentage bar can simplify the calculation when finding 35% of 500, by relating it to benchmark numbers 25% and 10%. This simplification process helped her in doing the calculation and giving the right solution of the problem. Before using help of the bar, Nouran, using her formal way, made mistake in the calculation. Even she correctly wrote down 35/100 x 500, she failed to come to the right answer. By using the bar, she also can visualize the problem and give clearer explanation to her friends. Although she admitted that the use of bar makes it easier to solve percentage problem, she does not prefer it. The reason is she already gets used to the formal way she often uses, even it usually leads her to the wrong answer.

Giving contextual problem asking for proportional comparison and letting them discuss the problem can help seventh grade pupils learn to understand using fractions and percentages as means for proportional comparison. As happening in the second lesson, even Nouran did not see the problem as a relative comparison one, the other two pupils, Andy and Jiwon did. Andy and Jiwon used fraction to solve the problem. Asked to use percentage to order all children based on their scores, all three pupils including Nouran could find all the percentages by their own way. At the end of the second lessons, Nouran expressed her understanding that both percentages and fractions can be used to solve a relative comparison problem.

In order to support pupils in developing their understanding of percentages, we can start by giving a rich contextual percentage problem to the pupils. A rich contextual problem is a problem that can be imagined by the pupils and has many ways to be solved, so the pupils can solve it in their own way and level. Introducing models such as percentage bar to the pupils is also helpful. The pupils can use the model to help them in doing calculation and visualizing the problem. They can also use the model as a tool to reason with while expressing their idea about the problem. Giving the pupils time to discuss the problem and to question one another’s solutions is very important in supporting their understanding of percentages.

According to the conclusion already drawn, some suggestions for improving mathematics education can be given. The first one is always start the lesson by giving rich contextual problem. Second, do not directly teach algorithms to the pupils, let them explore the contexts and find their own way to solve the problem. Emphasize also to the pupils that the way to solve the problem is as important as the solution of the problem. Third, always give opportunity to the pupils to work with model that can help them in developing their understanding of mathematics. The last one is always give time to the pupils to do discussion and to question one another’s solutions, because by doing these activities, they together can develop their understanding of mathematics.

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