Examensarbete
Learning mathematics -
how norms and a second
language may affect the
understanding of subtraction
with borrowing
A study in some classes in Kenya
Author: Susanne Erlandsson Supervisor: Berit Roos-Johansson Examiner: Jeppe Skott
Date. VT-HT15
Abstract
The purpose of this study is to observe some factors that may affect the understanding of subtraction with borrowing. The study is done in a foreign environment, in Kenya. Factors that will be looked into are: the classroom environment, the situation of learning in a second language. The study will also observe factors that can cause an erroneous answer and what those may look like. Within this area manipulatives will be mentioned.
The study has used a qualitative as well as quantitative approach. The qualitative method has been accomplished through interviews and observations, the quantitative method through tests given to the learners. For the analysis of the observations, Cobb’s and Yackel’s model (1995) of the mathematical classroom has been used. The study is interpreted from a sociocultural perspective focusing interaction.
The result in this study shows that the interaction in the classroom is important to the individual learning, perceptions of mathematics and the expectations on the individual. Learning in a second language can be a barrier. The use of manipulatives can work as a scaffold, but it can also hinder the learner to develop a deeper understanding.
Keywords: finger counting, influencing factors, interaction, Kenya, second language, sociocultural perspective, subtraction with borrowing.
Thanks
This study was made possible through the “Linnaeus Palme” scholarship. I am grateful for the opportunity to carry out my study in Kenya. To the people who supported me, when this only was a dream, and to those who made all this possible. In Kenya I would like to thank my mentors and the university, who made me feel welcome. I would like to thank the Primary School and the head teacher. Thanks to the teachers and learners that participated in this study. Thanks to Berit Roos-Johansson, who helped me in this process and gave me valuable feedback.
Abstrakt
Syftet med studien är att studera faktorer som kan påverka förståelsen av subtraktion med växling. Studien är gjord i en annan lärmiljö, i Kenya.
Faktorer som kommer att uppmärksammas är lärmiljön i klassrummet och att inlärningen sker på ett för eleven andra språk. Studien kommer också att uppmärksamma faktorer som kan bidra till ett felaktigt svar och hur de kan se ut. Inom detta område kommer laborativt material nämnas.
Studien har både ett kvalitativt och kvantitativt tillvägagångssätt. Den kvalitativa metoden har genomförts genom intervjuer och observationer, den kvantitativa metoden med hjälp av elevtester. Cobbs och Yackels (1995) modell över matematik klassrummet har använts som analysmaterial. Studien tolkas utifrån ett sociokulturellt perspektiv. Focus är på interaktionen.
Resultaten i studien visar att interaktionen I klassrummet är viktigt för det individuella lärandet, uppfattningen om matematik och förväntningar på individen. Undervisning på ett andra språk kan bli ett hinder. Användandet av laborativt material kan fungera som ett stöd, men kan också hindra utvecklandet av en djupare förståelse.
Nyckelord: andra språk, interaktion, Kenya, påverkansfaktorer, räkna på fingrarna, sociokulturellt perspektiv, subtraktion med växling.
Tack
Den här studien möjliggjordes genom “Linnaeus Palme” stipendiumet. Jag är tacksam för möjligheten gå genomföra min studie i Kenya. Till dem som stöttade mig när detta bara var en dröm och till dem som gjorde det möjligt. I Kenya vill jag tacka mina mentorer och universitetet, som välkomnade mig.Jag vill tacka Primary School och rektorn. Tack till lärarna och eleverna som deltog i studien. Tack till Berit Roos-Johansson, som har hjälpt mig i den här processen och gav mig värdefull feedback.
Table of contents
Abstract ___________________________________________________________________ i Abstrakt __________________________________________________________________ ii
1 Introduction ________________________________________1
1.1 Background __________________________________________________________ 2 1.1.1 The Kenyan school system ______________________________________________ 2 1.1.2 Subtraction in Kenya __________________________________________________ 4 1.1.3 Subtraction __________________________________________________________ 7
2 Aim _____________________________________________10
2.1 Aim __________________________________________________________________ 10 2.2 Research questions _____________________________________________________ 10
3 Theoretical background ______________________________11
3.1 Learning environment __________________________________________________ 11 3.2 Learning in a second language____________________________________________ 13 3.3 Subtraction and obstacles________________________________________________ 14 3.3.1 Manipulatives _______________________________________________________ 15
4 Method ___________________________________________18
4.1 Choice of method and datacollection method. _______________________________ 18 4.2 Selection ______________________________________________________________ 19 4.3 Implementation and analysis _____________________________________________ 19 4.4 Ethical considerations ___________________________________________________ 20 4.5 Validity and reliability __________________________________________________ 21 4.6 Collected data for further understanding___________________________________ 21
5 Results and Analysis ________________________________22
5.1 What standards governing what happens in the classroom and how may it affect the individual student's learning? _______________________________________________ 22
5.1.1 Classroom social norms _______________________________________________ 22 5.1.2 Beliefs about own role, others’ roles, and the general nature of mathematical activity in school __________________________________________________________________ 23 5.1.3 Sociomathematical norms ______________________________________________ 25 5.1.4 Mathematical beliefs and values _________________________________________ 27 5.1.5 Classroom mathematical practices _______________________________________ 28 5.1.6 Mathematical conceptions _____________________________________________ 29 5.2 How can understanding be affected when taught in a second language? _________ 29 5.2.1 The language of counting words ________________________________________ 30
5.3 What strategies do learners use which failed to solve the tasks correctly? ________ 31 5.3.1 Underlying causes of erroneous strategies _________________________________ 31 5.3.2 The strategies _______________________________________________________ 36 5.3.3 Some of the learners’ thinking and strategies when solving some tasks __________ 36 5.3.4 The strategy of using manipulatives – fingers, lines or circles. _________________ 42
6 Discussion ________________________________________47
6.1Method discussion ______________________________________________________ 47 6.1.1 The questionnaire ____________________________________________________ 47 6.1.2 Observations ________________________________________________________ 47 6.1.3 Find out the strategies with help of tests __________________________________ 48 6.1.4 Interviews with the learners ____________________________________________ 48 6.1.5 Language __________________________________________________________ 49 6.2 Result discussion _____________________________________________________ 49 6.2.1 Find out the strategies with help of tests and interviews ___________________ 51 6.3 Further research _____________________________________________________ 52 References _______________________________________________________________ 54 Appendix 1 _________________________________________________________________ i Appendix 2 ________________________________________________________________ ii Appendix 3 ________________________________________________________________ iii Appendix 4 ________________________________________________________________ iv Appendix 5 ________________________________________________________________ v Appendix 6 _______________________________________________________________ viii Appendix 7 ________________________________________________________________ x Appendix 8 _______________________________________________________________ xii
1 Introduction
No human being is an isolated island. In different ways we are affected by what we carry with us and the things around us. We are influenced and influence our environment all the time. In a classroom there are learners with diverse needs and various abilities. The teacher wishes to meet their needs and give them the best prerequisite for learning. The special teacher needs to be able to support and advise other educators and participate in the designing of the individual education plan (IEP). To be able to do this, awareness of factors that may influence the learning is vital.
What factors could be eliminated in order to ease the workload for learners with special educational needs (SEM – students1) and are there factors that, when educators use them, create SEM-students?
Learning is affected by context, how the interaction is done and what material used. The learner constructs her or his mathematical concepts and methods out of this context (Skott, Jess, Hansen
& Lundin, 2010). This study will show diverse factors that may affect the understanding of mathematics, specifically subtraction with borrowing. Addition, subtraction, multiplication and division are the four elementary, mathematical operations of arithmetic2 which is a base for further counting. Subtraction can cause troubles (Fiori & Zuccheri, 2005). Subtraction with borrowing contains several steps.
There are several factors that influence the learning. In this study they have been divided into three groups: 1. Factors from the outside. It could be home background, another mother tongue, traditions and so on. These are factors that influence the learning but it has not its roots in the school environment. This study will look into learning in a second language. The tradition is to some extent addressed. 2. Factors in school – the learning environment and cultural conditions.
How is the social interaction carried out? Which social norms are applied? What is allowed and how is the teacher and the learners expected to behave? Which are the sociomathematical norms?
Is the context including or excluding? 3. Factors that affect individually. How can learning be supported? What misunderstandings can occur? These are only a few of the factors that affect learning but the chosen that will be looked into.
Several of the points listed above affect learning overall not only the mathematics. Our goal is to meet every learner’s needs in every subject. By being more aware about what is going on, I (and others), have an opportunity to change our teaching for the learner’s best. On the other hand, there are factors that cannot be affected as home background, but an awareness of them can help the learner. I, as an educator, can also choose not to be influenced by these facts and instead add the problem solely of the learner.
This study is carried out in another country and another culture, in Kenya. There are two reasons why. I have been working in the Swedish school system for a long time. One reason is by facing another culture an awareness of other ways of teaching can be possible. There are several roads leading to the same goal (Thurén, 2007) and some are better than others or suit some learners better. The other reason is that this is an opportunity to have a new view of the Swedish
1 The expression, SEM-pupils, is mentioned by Engström and Magne, 2006.
2 The study of numbers, the properties of them in the operations
education, getting a new approach. This is an opportunity to discover phenomena in the Swedish education, which has been taken as self-evident by me. The teachers in Kenya follow their syllabus and have a system where results are emphasized rather than the way to the results. The success of a school is shown in their results from the exams. This influences the teaching.
Results are also accentuated in the Swedish school system. The tests show how well the schools have succeded.
The choice of subtraction, and subtraction with borrowing in particular, came out from a lecture.
During the lecture we were told that the learners in Kenya found subtraction difficult. When studying the results from the Kenyan test3 in mathematics, subtraction was named as an area where there were difficulties.
1.1 Background
Some of the words in this study are common used in Kenya. Learner is the same as students, standard is grade. All data is collected in Kenya.
Kenya is approximately the same size as Sweden with a population of 40 million people. There are 42 recognized languages that are spoken and there are two national languages, Kiswahili and English. How people live, how the school attendance works and how well you can speak the national languages depend on where in Kenya you live.
1.1.1 The Kenyan school system
Until 1963 Kenya was an English colony and the Kenyan school system has many similarities with the English one. The system was changed in 1985 and became an 8-4-4 school system, Primary-Secondary-University Kenya Certificate. The learners attend eight years in Primary School, four years in Secondary School and four years at the University. Primary School is free of charge since January 2003, but there are other costs such as buying school uniforms, activities in the afternoon and so on. How long the individual stays in the school system differs and many times it depends on where in the country the learner lives. In some places the learners only attend Primary School. In other areas it is more common that boys go to school but girls stay at home when they become older. 82 % attended Primary School in 2009 and more than 50 % continued to Secondary School (Nationalencyclopedia). The transition rate from Secondary School to University was 26.9% in 2007 (Kenya National Examinations Council 2012)
Learners in Kenyan school sometimes have to handle up to three languages. When learners start Primary school all teaching is in Kiswahili and English and further on only in English (except the lessons in Kiswahili). The textbooks are in English4. In the school where the study was carried out many of the learners speak English as their mother tongue but not all of them. The learners interviewed have the following mother tongues: English 23%, Kiswahili 65% and Kikuju 12%.
Some of the learners do not understand questions asked in English even when they are rephrased5. The textbooks are provided by the school. At the school collected data, there were
3 Annual learing assessment report, “Are our children learning?” Uwezo Kenya 2012
4 Except when they are learning Kiswahili.
5 Became apparent at the interviews.
not textbooks to every learner. The learners had to share the books and sometimes there were a whole group (up to eight learners) without any textbook. Problems with furniture, many learners in a classroom, lack of books and teaching and learning resources are some of the barriers of learning and teaching according to Aro, Namangala, February, Kalima and Koponen (2011). The school was in an ”ordinary” area, the pupils didn´t come from academic or rich families but they were not from the slums neither.
Lessons and exams
How to teach follows a pattern. It is taught at the university to training teachers. In the first five to ten minutes there is an introduction. In the observed lessons, it was usually the teacher showed how to solve the tasks in today’s work. After that, the learners worked in their textbooks for about 20 minutes and then a short follow-up. This pattern is taught to future teachers at the University but they are also told that they can fill the part in the middle with various tasks6. The learners are supposed to say some parts after the teacher, repeat or complete sentences.
Sometimes one learner answers the question. It is how the teacher intonates that shows if the whole class should answer in chorus or a single one should answer (Ponefact & Hardman. 2005).
That teaching strategy is very common. The “closed” questions dominate (Ackers & Hardman, 2001). The amount of lessons the learners are supposed to have and the length of them are defined. In “lower” Primary School (standard one to standard three) every lesson is 35 minutes.
In “upper” Primary School (standard four to standard eight) the lessons are 40 minutes. In the school visited, the teachers taught some subjects and were responsible for these in more than one class. So there were several teachers involved in one class at least from standard four. The syllabus addresses the objectives that shall be reviewed in the different standards. It is quite detailed in terms of the different elements and increase in severity.
The school year consists of three semesters. Every semester has got two national exams, midterm exam and end term exam. How well learners succeed on these exams, determine their grades and the success of the school and its rating is determined by these tests. The subjects that are tested are English, Kiswahili, mathematics, social studies and science. The weeks between midterm exams and end term exams are partly used to rehearse and repair what the learners’ have not grasped or misunderstood. It is possible to change between schools at any time. The schools cannot refuse to receive any new learner. Therefore teachers never know how many learners there will be in the class during the term. If the results of the national exams are good for one school, several learners can transfer to that school during the term. It does not matter if there are 50 learners in that class, the teacher has to give room for the new ones. The ages of the learners in one class can vary. In one of the classes in standard 4 the youngest was 8 and the oldest 11.
The annual learning assessment (ALA)7 indicates that boys tend to do better in mathematics than girls. (Girls do better in reading.) The results from the tests are lower in the dry areas of the country. It also says that the education of the mother influences the result. In standard 8 the learners have a final exam, Kenya Certificate of Primary Education examination. Finishing Secondary School there is a 12 letter grade system.
The Kenyan school “tomorrow”
6 Information from Dr Ulanga, teaching future teachers at the University in Kenya.
7 A test done nationally every year.
The school is part of a culture and its environment. Insights and activity patterns are built up historically. It shapes society where participation occurs through interaction with other people.
The culture also affects teaching, how the learner and the prerequisites are interpreted (Säljö, 2000). The Kenyan school system has an authoritarian structure. Tabuwala (2003) says the system in a country lies in its political and ideological nature. He sees a relation between politics and the school system in a country. He is an opponent of the “Westernization” of the schools in Kenya. That learner-centred pedagogy has its roots in the democratic tendencies and he says that this way of teaching will break the authoritarian system. Instead he wants culturally responsible teachers to lead the Kenyan school transformation. They should recognize the indigenous way of looking at knowledge.
1.1.2 Subtraction in Kenya
In the Swedish model borrowing is done and showed by drawing a line across the digit borrowed from. In the Kenyan model it is done the same way but besides that, it is also written above the digit how much is left. In the Swedish model the borrowed ten is written above the digit which needed it, as ten ones. In Kenyan schools the number one is written in front of the digit. The number one becomes tens combined with the other digit. Similarities within the two models, and mistakes done, are when borrowing is forgotten and instead calculating from the original number or borrowing from the wrong place value.
Calculating 37 – 19:
The Swedish model The Kenyan model
10 3 7
- 1 9 1 8
2 317 - 1 9 1 8
An example of the Kenyan algorithm in the classroom
Here is an example of a solution of the subtraction 351 – 125 (from a lesson when the teacher started up todays work in standard 4):
351 -125
”One take away five, it is impossible (or it is not enough). I borrow one from five, it becomes 11”
(Cross the tens with a line and the borrowed tens writes as a digit one in front of the one ones) 3511
-125
Goes back to the tens: ” It remains four.”
(Writes what is left of the tens, four, over the old tens)
4 3511 -125
”Eleven take away five is six”
4 3511 -125 6
“Four take away two is two”
4 3511 -125 2 6
“Three take away one is two”
4 3511 -125 2 2 6
Thinking when subtract in the Kenyan classroom
Observing lessons, subtraction has been explained as ”taking away” something. This observation is from standard 3 but it was common in the other classrooms as well. Looking at 7 – 3, it started with making 7 lines.
│ │ │ │ │ │ │
You are going to take away 3 so you mark three lines. The teacher could start from the left or the right.
│ │ │ │ │ │ │
After that you mark the lines which are left, in this case from the bottom.
│ │ │ │ │ │ │
The lines left are counted. Sometimes they drew circles but the circles were more used in addition. The learners used lines, circles or fingers when they solved the tasks.
Other numerical base systems
The learners learned to count with other numerical base system than ten when they are in standard 2. It is applied when they solved problems with time and weeks8. Having two weeks and three days and take away 5 days, they solved it in the following way:
Week Day
2 3
- 5
It is impossible to take away five days from three days. The learner has to borrow from the weeks. When borrowing one week it means getting seven days. So in the middle of the operation another numerical base system than ten is used temporary.
Week Day
1 2 7 (+)3 = 10
- 5
The “week” goes down by one. The learner adds the days and gets ten days. Now is it possible to do the operation. Ten days – five days = five days. In the column for weeks there is one week left.
Solving problems with time ( numerical base of 60) and length (numerical base of 100) is solved in the same way.
Subtraction in the Kenyan syllabus (Ministry of education 2002) In the beginning of the syllabus there are overall goals and guidelines.
Subtraction is reviewed in the following way:
Standard 1: Subtract numbers not exceeding 99 vertically and horizontally without borrowing, identify relationship between addition and subtraction, subtraction as taking away, subtraction of 1-digit numbers from 2-digit numbers based on basic additions facts, subtraction of 1-digit numbers from 2-digit numbers without borrowing, subtraction of multiples of 10 (page 5).
Note, for every addition fact here are two subtraction facts related to it: 5+3=8, 8-3=5, 8-5=3 Standard 2: Subtract up to 3-digit number from up to a 3-digit number without borrowing, subtractions involving missing numbers (page 7).
8 One of the questions was what topics they found difficult to teach in mathematics. Some of them answered time and the days of the week. Going back and asked these teachers why they thought time was difficult to teach. They told me how they calculated with other numerical base systems when they worked with time and that was sometimes difficult for the learners to grasp.
Week Day
1 2 7 (+)3 = 10
- 5
1 5
Standard 3: Subtract up to 4-digit numbers from up to 4-digit numbers without and with borrowing (page 9).
Standard 4: Subtract up to 5-digit numbers from up to a 5-digit number without and with borrowing (page 13).
Standard 5: Work out addition and subtraction involving up to 6-digit numbers (page 17).
Standard 6: Addition and subtraction of numbers (page 22).
Standard 7: Addition, subtraction and multiplication involving whole numbers (page 27).
Standard 8: Work out problems involving operations on whole numbers (page 33).
1.1.3 Subtraction
Different facts about subtraction will be displayed here. Since this work has the algorithm in focus other methods will not be reported.
Review of subtraction
Subtraction is one of the four mathematical operations of arithmetic9. Subtractions can be used to describe the difference between two numbers, a comparison. It is also used to tell what is left after a removal “taking away”. The Nationalencyklpedia gives the following explanation;
“Subtraction, subtractio, “evasion” from the Latin subtraho “pull away, remote”.
The algorithm, necessary knowledge
An algorithm could be explained as a procedure to solve a problem. The procedure has several steps that have to be carried out in the right order. They follow a pattern. Subtraction is anticommutative, the operation must be carried out in the order it is standing, the terms cannot be reversed (in addition it doesn´t matter what term, summand10, to start with). The problem is solved vertically, by writing the subtrahend under the minuend11. The subtraction algorithm is complicated. In order for the learner to understand, explanations are required. In the top line, the minuend12, it is allowed to rearrange the digits if needed. It is also allowed to write two digits in one square. On the contrary the value must be unchanged (McIntosh, 2010). There are several basic elements to solve a vertical written subtraction algorithm. The learner must have prerequisite skills such as number knowledge (the number line forward and backward, preferable automatic, and tens transition), the place value system, how to write the algorithm and how to perform it.
These steps can be carried out mechanically, an ”instrumental understanding” (Skemp, 2006).
The steps are carried out by rote, orderly, without understanding why it is done. Unlike
“relational understanding” (a.a.), which is about knowing what to do and why. When a learner uses ”instrumental understanding” (further it will be referred to mechanically understanding) no number knowledge13 (basic numerical facts) is needed. It neither has to be automatic. However, the learner must have knowledge of the relationship between the figure and the number and how
9 The others are addition, division and multiplication.
10 The summand is what you add together in addition
11 The minuend is what you start with and the subtrahend is what you take away.
12 In the operation 8-4, 8 is the minuend.
13 Here it refers to the addition- and subtraction table up to 20
to think when it is subtraction. The knowledge consists of two different components (Löwing&
Kilborn, 2003). One competence is comparable to understanding, which in this case comes to understanding the operation. The other is the skill that is needed to perform the operation.
Known causes of ”incorrect answers"
With mechanically understanding it is possible to solve the task as a rehearsed ritual. No need to understand the arrangement of the ritual is necessary. If there is any misconception about any steps or any lack of prerequisite skills, it can lead to an incorrect answer. Under the circumstances it is difficult for the learner to see where it went wrong or find other ways to solve the task. It is possible that the learner do not react to the incorrect answer. Because of the lack of previous knowledge, estimation is undoable (Löwing & Kilborn, 2003). Other main reasons to incorrect answers could be: Mixing the additions- and subtraction algorithm, writing the numbers in the wrong positional number system, calculating with ”always subtract the smaller digit from the larger”, do not recognize that the answer is unreasonable and miscalculating (McIntosh, 2010). Different reasons why mathematics becomes difficult has been listed by Aro, Namangala, February, Kalima and Koponen (2011). They have written a report concerning African schools.
Some of the reasons according to the report are the teaching language, learners' difficulties with numerical abilities and memory problems. They also address teacher education and training.
Always subtract the smaller digit from the larger
The learner reverses the minuend and subtrahend14 when the subtrahend is larger than the minuend. For example: 3-7 instead becomes 7-3 when it is not possible to take 7 from 3. A different kind of thinking can occur if the minuend is 0. For example in 0-5, the answer can be 0, nothing can be removed so the answer is 0. When learners do not have the understanding but follows the routine to solve a task, these errors tends to occur (Anghileri, 2006).
Errors in the sequential operations in the algorithm
A sequential operation refers to, for example, the subtractions of tens in a problem where hundreds, or more, is included. Errors made in the sequential operations may be due to several causes: 1.”Forgetting” what kind of operation it is and start counting addition in one of the following steps. 2. Forget that borrowing has been done in the step earlier. Forget to show that a borrowing has taken place. 3. Errors can occur when it is zero tens and borrowing has to take place from the hundreds. The learner borrows ”10” directly to the ones but forgets to shifts to the tens. Sometimes borrowing can be done from “zero” though it is impossible. 4. Borrowing from the hundreds though it is enough tens. Sometimes the borrowing is written directly over the ones but can also be written over the tens. When written over the tens, the result of this can be that the subtraction of the tens could be for example 13 – 2. The answer 11 is written under the tens or it is split into 1 hundreds. (If the learner should remember to add the extra hundreds when calculating the answer could be correct.) 5. Going directly to the hundreds if borrowing is necessary and borrow it to the ones, if there are not enough tens in the next step, another hundreds is borrowed.
14 In the operation 3-7, 3 is the minuend and 7 the subtrahend
Miscalculating
Through different strategies used by the learner, the answer can be one more or one less than the correct answer (Johansson, 2011). The reason could be finger-counting. There are also more types of miscalculating.
2 Aim
2.1 Aim
The aim of this study is to reveal factors that may influence the understanding of subtraction with borrowing.
2.2 Research questions
What standards are governing the classroom?
How can understanding be affected when taught in a second language?
What strategies do learners use, in the written subtraction algorithm, which failed to solve the tasks correctly?
3 Theoretical background
Several areas from previous research will be looked in to. The basic theory chosen is about learning environment in the classroom. The study is interpreted from a sociocultural perspective and a constructivist point of view, a socioconstructivistic theory. Learning depends upon the social structure and the interaction in the classroom. It cannot take place as isolated phenomena (Dysthe, 2003). In subtraction with borrowing different sources will point out known difficulties.
Another area is teaching in a second language.
According to Piaget’s constructivism theory, learning happens through assimilation. It starts with the individual and goes to the social. The knowledge is individual and creates inside their mind.
New facts incorporate with the old knowledge and the result is new knowledge, but the knowledge can differ between individuals. Knowledge cannot be discovered only organized. It happens through two steps. First assimilation, adjust to new conditions (building a frame) and after that accommodation, adjust to a new approach. The individual has a drift to learn and that follows certain steps. The surroundings can create conditions for learning. The individuals can discuss known and new facts and reach a new understanding but their new understanding is individual, building on their previous understanding. Their new understanding could be totally different from each other. It is important for the individual to experience the sharing of knowledge and happens through interaction. The language is proof of the individual understanding. If any sharing took place cannot be determined (Skott, Jess, Hansen & Lundin, 2010).
According to Vygotski’s sociocultural theory, learning happens through participation. It starts with the social and goes to the individual. In the interaction new knowledge can be created that could not be learned on your own. The teacher has an important role to guide and ask questions in this learning process. The context is influenced by the culture and society and its means of expression is the language. Learning is possible through the zone of proximal development (Skott, Jess, Hansen & Lundin, 2010).
Socioconstructivism sees learning as both acquisition and participation. The two sides cannot be joined as one theory but both ways is necessary to explain what is going on in a classroom environment. In the classroom there is not only one individual but a whole class. Something happens in the room when there are many participants and other rules or norms form the standards or the teaching and learning. There are a variety of factors that influence the process.
Knowledge also has a cultural component (a.a.). In Cobb´s and Yackel´s model (1995) there are both a social and a psychological perspective, there is a reflexive movement between these perspectives. They influence each other. The language in the classroom also reflects both sides.
The language is an expression of the thinking and at the same time a possibility to be able to think.
3.1 Learning environment
The starting point is Cobb’s and Yackel’s (1995) model of what happens in the classroom during a mathematic lesson. The model is interpreted by Cobb’s and Yackel’s (1995) report and by Skott’s, Jess’, Hansen’s and Lundin’s (2010) interpreting of the model. Yackel and Cobb (1996) argue that a constructivist perspective is not enough to understand what happens in a classroom, and how learning occurs. Social aspects must also be included in the mathematical activity (a.a.).
Learning is both an individual process and assimilation to the culture. Their theoretical perspectives are constructivism, symbolic interactionism and ethnomethodology. Symbolic interactionism means striving to something meaningful, it is done through social interaction and can be modified in the reflexive process. It can only take place in the context of the individual (Handberg, Thorne, Midtgaard, Nielsen & Lomborg, 2015). In ethnomethodology interaction is fundamental. In the interaction the participants defines, pursues and reach their goals. The social norms are a result of the interactions (Nationalencyclopedia).
In every classroom there are different norms governing what is said and done. They can be general or linked to the topic. This phenomenon was called “The hidden curriculum” (Broady, 2007). When learners participate in this interaction in the classroom, they learn the code, and also learn what roles everyone has. Through this they also learn how mathematics works in school. What is a good question, a good solution, what form of response is an acceptable response, etcetera. In a classroom, a class together with the teacher come up with some "truths"
of mathematics, which in this context is taken for granted and do not need to be explained or questioned. It is important to point out, that it not only depends on the teacher what these "rules"
looks like, but it is to some extent an interaction between the learners and the teacher.
Level Social perspective Psychological perspecitve
A 1. Classroom social norms 2.Beliefs about own role, others’ roles, and the general nature of mathematical activity in school
B 3. Sociomathematical norms 4. Mathematical beliefs and values C 5. Classroom mathematical practices 6. Mathematical conceptions
(Cobb’s and Yackel’s model of the mathematic classroom, 1995:6. I have added the levels A, B and C, and level 1 to 6, to the chart.)
The theory contains three levels which are about the norms and beliefs that control the mathematics lessons. The first level is about how they govern the classroom life; what is said, what is allowed and so on. The next level is about how they are linked to the subject of mathematics. The last level is about what skills and concepts the learners develop. It is also divided into two perspectives.
1. Classroom social norms
This applies to both teachers and learners. These norms can be general or related to the topic.
The general social norms play a big role in the learning process itself and how it develops. These standards, which can be hidden or implied, are important that learners learn (Broady, 2007).
2. Beliefs about own role, others’ roles, and the general nature of mathematical activity in school In the social interaction between teacher - learner and learner - learner, there are norms that all are expected to act upon. These norms are learned in the actual interaction. They govern the notion of the learner and others on how they are expected to act.
3. Sociomathematical norms
These norms govern what mathematics is, what is good mathematics and what characterizes a good mathematical explanation. These socio-mathematical norms develop in the interaction. It is
not just the teacher's ideas that govern; they may be different in different classes. The answers that are accepted are governed by the answers learners will deliver
4. Mathematical beliefs and values
The learners' own conceptions of mathematics can vary, and may be the capacity to quickly and correctly be able to say the answer or developing different strategies to arrive at the answer. This means that learners may expect that there is always one way and one answer to a problem. The opposite is when they get surprised if everybody has done it in the same way and arrived at the same answer. The belief may also be that mathematics is something that is done in the book or something that is done with others.They can see mathematics as something only done in school and which do not have any connection with daily life. These performances are controlled by the socio-mathematical norms while these beliefs help to shape them. These norms provide the picture of what mathematics is in a class.
5. Classroom mathematical practices
The third level is about the subject of mathematics. Some parts or methods have gradually been accepted by the class as truths and they need no longer be argued for, but have become of general use.
6. Mathematical conceptions
Although there is a generally accepted way, which does not mean that every single learner thinks alike. When the learner comes to a math lesson, he or she has expectations or beliefs about what will happen because of the experience of previous lessons. The learner has a template for how the teacher is expected to act, how he himself or she herself should act and how the other learners are expected to act. This expectation can be positive (for example engaged in work) or negative (no commitment are to be shown). Learners learn what the teacher expects and "guess"
sometimes what answer the teacher wants without perhaps having the line of thought clear (Skott, Jess, Hansen & Lundin, 2010).
The teacher has got the role as a leader in a classroom. It is not only about teaching. The teacher conveys the values and standards the society stands for. They should nurture learners to become good citizens. The teacher is the authority (Thornberg, 2013). Authority is explained as: “the exercise of legitimate influence by one social actor over another” (Encyclopædia Britannica).
Through this authority the teacher has got the power (Ackers & Hardman, 2001). In order to make it possible to exercise this power it has to be recognized by all participants. It has to be a relationship and an interaction between the leader and the group (Thornberg, 2013). The expectations and interactions are based on the teacher’s and learner’s perceptions of each other, their roles (Hundeide, 2003).
3.2 Learning in a second language
There are at least 42 different tribe languages spoken in Kenya. When the learners come to school, they get approached with a school language. The learners who do not master the language need to learn the new language and at the same time the concepts in the school language. Multilanguage learners usually have different lexicon in different languages as a result of using them in different contexts. Learners in this situation lack meaningful words in their basic vocabulary (Salameh, 2012). To learn mathematics in a second language faces challenges
on multiple levels. It is a challenge to learn mathematics at the same time as learning the language spoken in the classroom (Barwell, 2014). There is a difference between everyday language and mathematic language (Sterner & Lundberg, 2002). It is important to know the mathematic language. It is a source of power (Fredrickson & Cline, 2009).
The language at school could function as a barrier instead as a tool for communicating (Brock- Utne, 2007). A result of that could be that learners achieve little during the lesson. The learners need help to translate the school language to their mother tongue (Verzosa & Mulligan, 2013).
The teacher can be an important connecting link. (Barwell, 2014). In Kenya the teachers can be positioned in an area where another mother tongue is spoken. Then the teacher cannot speak the learners’ mother tongue which complicates the possibility to support the learners or being a connecting link. When something is not understandable, the learners could try to find
“keywords” that can help them to understand (Verzosa & Mulligan, 2013). Errors can occur from misunderstandings and the learner’s own way to get through the situation. The task can be understandable but the number line is not known automatically on the teaching language. The result can be that energy taking strategies are used, to be able to reach an answer. The learner can
“get lost on the way” and the answer turns out to be something else but the right one. They also try to “read” the teacher and answer the answers they think the teacher wants (Leron & Hazzan, 1997). Another obstacle could be the mathematic language. In mathematics the same words are used as in daily speaking, but they have a different meaning. For example borrow in subtraction and in the daily life borrow (Chinn, 2004).
3.3 Subtraction and obstacles
A vertically written subtraction algorithm is done from the right to the left. That is different from the first years in school when the task is written horizontally. The ones’ digits and tens’ digit can be counted separately, split tens15. In those cases the calculation is done by starting from the left.
The most common mistakes16 are borrowing and ”always subtract the smaller digit from the larger” (reversing the minuend and subtrahend) (Johansson, 2011). The operations that are done in a written algorithm, results in that the learner needs to master ”the big subtraction table”17 Some of the borrowing mistakes are different in Kenya compared to Sweden.
Strategies are used in the following way; counting all → counting from the first → splitting tens
→ knowing basic numerical facts (Johansson, 2011). Counting all is when the learner first count up to the minuend and after that takes away the required number by counting backwards.
Counting from the first is solving the task by counting backwards one step at a time. Splitting tens is for example 13 - 6, splitting the subtrahend into 3 + 3, go three steps back to ten and then additionally three steps backwards to seven. Knowing basic numerical facts is when the learner
”knows” that 13 – 6 is 7 (a.a.). These steps show development of counting strategies. A research was done including learners who were supposed to be in need. They solved more tasks when they used manipulatives than when they tried to solve it with mental arithmetic. The strategy
”counting everyone” was used, and counting from the first. This method often gave the incorrect answer (Johansson, 2011). The reason could be that the learner does not master the strategy,
15 52 + 31 splits into 50+2 and 30+1, you count the tens and the ones, 80+3 =83
16 In Sweden (Johansson 2011)
17 ”The big subtractions table” contents subtraction within the area 0-20. “The little subtractions table” is from 0-10.
which number the counting should start from. If the task is 13 – 5, should you go backwards from 13 or 12, and when five backwards steps is done, is it the answer or is it the one after that?
To count with fingers it is even more difficult. When they instead solved the task using mental arithmetic they finished fewer tasks but got more correct (a.a.). The SEM- learners and other learners differed in that other learners more often used more advanced strategies. In school SEM- learners are often offered manipulatives in order to ease the understanding for the operation. This research shows that there is a risk that the use of objects stops learner from developing strategies and instead they get stuck with troublesome and ineffective counting strategies (a.a.).
To comprehend the reasonableness in the answer can be difficult. The written vertical algorithm can contribute to this. There is a risk that the sum 645 is perceived as a series of digits standing beside each other; 6, 4 and 5.Working with place value in the algorithm does not necessarily mean that they are understood as meaningful sums (Johansson, 2011).
When it comes to subtraction, subtraction with borrowing is the most difficult one (Chinn, 2004).
Many mistakes are done when borrowing from “zero”. Another is when the subtrahend is bigger than the minuend (Fiori & Zucchini, 2005). The learner applies “Always subtract the smaller digit from the larger” – strategy. Fiori and Zucchini (2005) have done a large study to analyze which mistakes are done. They were divided into four groups; 1. Mistakes connected with the technic of borrowing. 2. No borrowing is done though it is needed. 3. Calculating errors like 7 – 5 = 3. 4. Mistakes that cannot be explain by logical mathematic but can be explained by the learner. It could be that addition has been done instead of subtraction. They noticed that
“calculating errors” were mixed with “procedure errors”. The mistakes did not need to appear at every task. They came to the conclusion, that the methodology the teacher used also reflected the results. They came to that conclusion when they tested several classes who had the same teacher.
3.3.1 Manipulatives
In the section above it was written that manipulatives could hinder the development of calculating strategies (Johansson, 2011) and that the learner instead could be stuck in a model. If not several methods are offered to calculate addition and subtraction, it is easy to only use one method. Usually counting forwards when it is addition and backwards when it is subtraction.
The fingers can be used as a tool. The learner easy reaches to one more or one less, than the correct answer. As a result of that, the learner does not know which number to start counting from in order to get the answer (McIntosh, 2010). Counting with fingers can hinder the learning process (Aro, Namangala, February, Kalima & Koponen, 2011).
McNeil, Uttal, Jarvin and Sternberg (2009) did a test with fourth- and sixth graders, an experiment with manipulatives. One group solved tasks with manipulatives and the other without. The results showed that the group with manipulatives did more errors than the group without materials when it came to calculating errors, but less conceptual errors. The test was carried out again, this time with fifth graders. They got better designed manipulatives. (The tasks were about money. The last group got material that more looked like real money compared to the first groups.) The group with better designed manipulatives did more errors than the control groups. Their conclusion is, that manipulatives is both an advantage and disadvantage. The more lifelike money was supposed to more connect the task with everyday life. After analyzing the last test they are not sure that the colourful and realistic money caught the attention of the learner in a
mathematically positive way. Manipulatives can be used with advantageously to make the problem make sense, according to them. They also indicate that it is possible to get stuck in the material instead of focusing on the mathematics.
Kamii, Lewis and Kirkland (2001) raise the question about if manipulatives are useful. Their basic thoughts have its origin in a model done by Piaget18 and they talk about logicomathemathical knowledge. Logicomathematical knowledge has got a mental relationship as a base that every individual creates insides themselves. For example it expresses itself at an observation of two flowers with different colours. The focus can be on the similarities or the differences. Inside, it is perceives that there are two of them, but the eyes don´t see the digit two, only two flowers. If the focus instead is on the colours of the flowers it is about physical knowledge. The physical knowledge is about object in the external word. This knowledge is also about the ability to estimate the weight of an object and that at swing stops swinging after a while. This knowledge has its base in experiences done in the external world. They mean that the logicomathematical knowledge has its source inside the child and the physical knowledge comes from the reality around the child.
In the young child these knowledges coexist as ”one knowledge” but when the child grows the logicomathematical knowledge separates itself and becomes independent. If we for example talk about three bananas it is physical knowledge because we are talking about bananas. When the learner faces 3 + 3 and can interpret it, the logicomathimatical knowledge is separated from the physical knowledge and can stand for itself. Furthermore they talk about different kinds of abstractions. One is called empirical abstraction (also called the simple abstraction) and the other one is called constructive abstraction (also called the reflective abstraction). The empirical abstraction sees for example the colours of the flowers. The constructive abstraction creates mental context such as “five”, “different” and so on. The abstraction also creates the digits and later on it makes a relationship between”2 + 2”. Taking this in consideration Kimii, Lewis and Kirkland (2001) mean that it shows that learners ”create” mathematics through constructive abstraction. They also mean that what we call “thinking” or “reasoning” come from this ability.
School often uses manipulatives or figures. To put the numbers in a context often leads to a better understanding of the situation. To only have 3 + 3 could be incomprehensible. By using materials, 3 pencils + 3 pencils, it could become meaningful and the addition is comprehensible.
Just by putting the problem into words could sometimes be enough. In this example the manipulatives works as a scaffolding, a good helper. The logicomathematical knowledge has not started to work on its own and needs help from the physical knowledge. The learner needs to develop the constructive abstraction. Manipulatives can sometimes prevent a development of thinking and strategies. To use different base-ten blocks when the value system is taught, addition with tens transitions or subtraction with borrowing is according to Kamii, Lewis and Kirkland (2001) not effective. They are not useful. They go on by saying that the learner can´t distinguish the object one in the ones and the number in the ones. It is the same with the tens.
They mean, that when a learner in standard 1 says 34 it means 34 ones unlike from an adult who thinks about 3 tens and 4 ones. The adult can think about tens and ones at the same time but the
18 Piaget (1971) had in an articel: Biology and knowledge explained the concepts of physics and logicomathematical knowledge,
learner alternates between the two expressions. One way to see this is by counting items in bowls. The bowls contents ten items each. The learner counts 10 – 20 – 30 and so on. The adult can see the context and counts 1 – 2 – 3, referring to tens.
The understanding of ”0” influences how learners can solve subtractions (Fiori & Zucchini, 2005). Learners start to experiment with ”0” in the positional number system with manipulatives but they do not move on to step two. Step two is turning over to abstract thinking. It takes practice to go from the concrete thinking to the abstract according to Fiori and Zucchini (2005).
The expression “Special educational needs in mathematics” shows a new way of thinking about the learners who sometimes struggle with mathematics. The expression means that the learner has needs and shifts the focus from the learner to the approach instead (Lunde, 2011). It could also be explained that these learners do not reach the objectives in mathematics and that they show it early in the education. It is the school that determines that the learners do not know mathematics and these learners cannot pass the grades (Engström and Mange, 2006). SEM- learners show weaknesses in processes that are usually automatic (Fredrickson & Cline, 2009).
4 Method
The approach of the research was qualitative as the purpose was to create a deeper understanding within an area of subtraction with borrowing. The focus was on the individual in the interpretation of results. Teachers and learners have been studied to see how they interacted and functioned in the classroom, how social connections influenced and made the individual learner involved in the context (Dahmström, 2011). Their views and opinions were revealed through interviews. It also has elements of an ethnographic survey. Regular observations have been carried out during a period of time, written information has been collected in order to develop an understanding of the culture of the group and their behavior (Bryman, 2002). The study has quantitative elements, in that measurable data has been collected. The quantitative examination form was cross-sectional (Bryman, 2002). An explaining focus, causality focus, has been sought.
One the other hand there could be several various connections that influenced the outcome in the classroom. The interpretation method has been hermeneutical though having some prior understanding. However, it was important to ensure that the factors that contributed to solving a task incorrectly, to a large extent cannot be transferred or generalized. Some of the factors might agree with previous studies and may strengthen that view. As for the erroneous solutions, they can be generalized and help interpret incorrect answers, that learners in other environments do.
Some of the erroneous solutions could only be interpreted in the Kenyan context, as they use a kind of algorithm, others can be made by learners everywhere.
I attended courses in teacher education at the University in Kenya. That gave some insight in what the education system looked like and how future teachers were trained. That background and studies of the Kenyan syllabus is an important tool included in my analysis.
4.1 Choice of method and datacollection method.
To be aware of the culture or the learning environment that prevailed in the classroom, a number of mathematic lessons were observed. My role was not to participate in the actual teaching, but to observe. It was an open observation. At the same time as the observations started, several teachers answered a questionnaire (appendix 1). My intention with the questionnaires was to gain an insight into how they perceived their situation as a teacher of mathematics and what strategies they used to meet all learners' needs. The aim was also to quickly get into the approach that prevailed there and gain a greater understanding of my surroundings. The seven teachers had taught from nine to approximately 30 years. Everyone but one teacher enjoyed teaching. They taught from standard 2 to standard 6.
After that, learners' skills were tested with two tests. First they got a test of addition and subtraction in the number range 10-19. It was made to ensure learners' skills in calculating in the number range which is relevant when solving subtraction with borrowing. After that a test about subtraction with borrowing followed. The tests that were used are the Swedish national agency for education tests, the “Diamondtest” (2013). The diagnoses have a theoretical didactic theory basic and are structured on the basis of areas. They consist of "naked numbers" and are easily corrected without doubts. The tests can assess whether learners have the knowledge required to solve a certain type of tasks. If a learner does one or several errors, a follow up, an interview is recommended. The test has got a time limit; if it is done in three to four minutes the learner master the tasks. Eight minutes is the time limit.
Using the results from the tests, a number of learners were chosen for an interview. The learners answered questions about their age, how they felt when they had mathematic, if they intended to go on studying, if they were aware of the mathematics they used at home in "everyday life" and so on. After that, they were asked to explain how they had solved certain tasks (appendix 2).
Once the interviews were completed and a further analysis had been made, a feedback was done to the teachers, whose learners were tested with the “subtraction with borrowing” -test. The intention was to give a feedback on what was discovered in the tests and interviews with the learners.
4.2 Selection
As Linnaeus’ Palme project cooperates with several universities, a Primary School that was situated on the university area was chosen. Classes from standard 2 to 5 were selected for the observations. The original plan was to observe from standard 1 to 6 because I teach in these standards. Different reasons made it impossible to have a meeting with the teachers. In order to make it possible to have a meeting and inform the teachers, I decided to involve a less number of teachers.
The choice was to test subtraction with borrowing and analyze the learners who were in standard 4. The reason for this was, that they learn subtraction with borrowing in standard 3. The first test was in mental arithmetic addition and subtraction in the number range 10-19 (appendix 3). It was chosen because it is within that area subtraction with borrowing takes place. This test was also made in standard 2 and standard 6 to see if there was a progression in learning. The second test included five tasks of subtraction with borrowing (appendix 4). This test was also made in standard 6 to use as a comparison. The tests were also made to see, if learners were in difficulty or if they just have difficulty with the algorithm of subtraction. If learners cannot solve the subtraction algorithm but have no problem with "calculating", that will show by many correct answers on the first test but several errors of the second one.
When the standard 4 tests were analyzed, some learners were chosen for an interview. The learners selected, were those whose answers that were not interpreted or understood. Some learners, who had made no mistakes or had made incorrect solutions that were understood, were also selected, but the main groups were from the solutions that not were understood. When all this was completed a feedback with the teachers individually was done.
4.3 Implementation and analysis
The head teacher was contacted first, a meeting with the staff was needed, before starting with any work. In the meeting I introduced myself, informed about the study and informed about the ethical rules for the study. Unfortunately, a strike started when arriving there which lasted for two weeks. A number of lessons in mathematics were observed. The observations were made in several different classes. Notes were taken by sitting at the back of the classroom with paper, a pen and a watch. It took a number of occasions before teachers were accustomed to my presence, and only then, a chance to observe the actual conditions were possible (Dahmström, 2011). In the classes observed, teachers filled out a questionnaire. Going through the answers immediately
made it possible to quickly return to the teachers if not understanding the answer, or a depth or explanatory responses was desired (appendix 1).
The tests were conducted in the classes. From these, 22 learners were chosen for the interview.
Totally more than 200 learners were tested with test one and about 130 with test two. A conscious subjective choice was made in order to get answers to the issue (Dahmström, 2011). A representative selection when it came to the incorrect solutions. The questions were not the main purpose, but some interesting information was revealed. They were used more as a "door opener”. The learners were interviewed one by one. The interview was supposed to take place in a private room (the library). My goal was to be alone with the learners, so they would feel comfortable to answer the questions. Several times teachers came in anyway and sat somewhere in the room, not to monitor what happened, but to rest for a while. This could obviously affect the learners negatively. In some interviews, we understood each other, but there were also learners, who did not understand what was said. In those cases, we almost immediately went to the part where they explained how they had solved the tasks.
4.4 Ethical considerations
I contacted the Ethics Council in Sweden to find out which regulations to apply. The answer was, that it could be different in different countries. After that a contact was made to those responsible for the scholarship. They said that the head teacher at the school already had seen to the permits needed. Before the study began, a briefing with the teachers at the school took place and they were informed of the four ethical main requirements: the information requirement, the consent requirement, the confidentiality and use requirement (Vetenskapsrådet, 2002).
It was stressed that all information would be anonymized. No learner or teacher would be able to be identified when the material is presented. It was important that they felt safe in my company.
Since coming as an "intruder" no expectations can be made about seeing the whole truth. The more they felt they could trust me, they were not to be pointed out or framed, the more of classroom reality was revealed.
The learners were not accustomed to be selected for interviews and hopefully they felt calmer if they were given some general questions to begin with.The first question was about how old they were, it was an interesting question in a way, because learners could start school at slightly different ages, but for them it was an easier question to answer and it was not challenging. It could be an advantage when coming from another context and had no prior relationship with them (Alderson & Morrow, 2011). When asked how they felt about mathematics, getting an honest answer is possible because of no teaching relationship with them. At the same time a consciousness had to be there that it might not be so. Learners could answer what they thought that the school wanted them to answer or answer what they thought I wanted to hear.
Expectations were that they would feel insecure to some degree in the interview situation.
Therefore, the interview started by asking if it was ok asking some questions. The issue was a way for me to show them respect and gave them an opportunity not to participate, although I did not think anyone would answer no to this question. I would not be unknown to them, being observing the class for a number of occasions, no explaining about me was needed. An awareness of my advantage in the meeting had to be recognized. It is important to make it as good as possible for the learner.
4.5 Validity and reliability
It could be questioned if the study was reliable in the observed classes. It is social and interactional data that have been collected. The data have been interpreted and is in a way always subjective. The study cannot be generalized to everywhere in Kenya, not even in all classes at the school. The tests would have high reliability and validity in the classes that were tested (Bryman, 2002).
4.6 Collected data for further understanding
Studies were made through observations. One purpose was to see how interaction worked in the classroom. It was also carried out to see how subtraction with borrowing was taught. It was made to make it possible to understand and make conclusions about factors that may influence the understanding (Bryman 2002). It also made it possible to analyze the incorrect answers that came up.
