Excerpt

## Least Common Multiples and Squares: Reflections on co teaching Mathematics in a community based school

**By: Asma Ahmad Mamsa**

## CONTEXT

This paper is based on my experiences of mathematics classroom teaching in grade five in a community based school in a developing locality in Karachi, Pakistan. The school was big in size and established in a proper school building. Majority of the children seemed to come from middle and lower middle SES^{1} backgrounds. The class in which I taught was grade five, section A, number of students in the class was about 25. Throughout my visit the management was very cooperative. The teaching learning environment of the school was pre dominantly traditional with my cooperative teacher Jaweria strictly adhering to jug and mug theory.

## CONTENT ANALYSIS

Despite being traditional the school was following Oxford University Press textbooks which showed it wanted it students to learn from the best. The unit of the book which I taught with Jaweria’s (pseudonym) help was of factors and multiples. She was the Mathematics teacher of that class. Factors and multiples belong to the strand of number and operations of arithmetic and follow the number theory. Factorization falls under the category of divisibility but a special type of divisibility, as Zazkis and Campbell (1996) say ‘thematized’ divisibility one which is associated with cognitive structures like factorization and prime decomposition. As it forms higher order thinking or schema building. Perhaps knowing divisibility can be referred to as procedural understanding for performing factorization and prime decomposition which require conceptual understanding (Sfard, 1991).

## STUDENT THINKING

### Least Common Multiple

During the session when I started teaching the students in grade 5 about Least Common Multiple they picked on well with the activities which made the meaning of ‘common’ and ‘common multiple’ clear. However, when it came to brainstorming about the term LCM as a whole, their responses about the meaning of the word ‘least’ surprised me. They said it meant *lamba ^{2},* long, latest and length; one student finally said less from which I built on to least. Initially, I felt the problem was of vocabulary and associating words. When I investigated further on this issue by conversing with students I found that the LCM had not been discussed in class as ‘Least Common Multiple’ it was always referred to as ‘LCM’. Earp and Tanner (1980) in their study of sixth graders found that only 50% of the students were able to articulate the meaning of mathematical terms. Moreover, that mathematical terms are inadequately discussed in class is also suggested by this study. This was the problem that LCM had not been properly discussed in class and the students did not possess sufficient vocabulary to make out the meaning of least. ‘Common’ and ‘multiple’ were frequently referred in class so students knew the meaning of those words. In order to overcome this barrier I think mathematics teachers should develop a grade wise collection of mathematical terms and they should be learnt by all students (Garbe, 1985).

### Square

When I had to co-teach the concept of square as an exponent I planned to give the concept by linking it with the area of the geometrical square. I started the lesson by asking them the definition of square, hoping that they would come up with the definition of geometric square as well. Surprisingly, the students initially only told me the definitions of the exponent square. On deep probing they finally told me the definition of the geometrical shape as well. I asked them if there is something similar between the geometric square and exponent square. They told me no, the common response for geometric square *was" woh shape haijo geometry mein hota hai”* (it is a shape which is present in geometry) and for exponent square was *“woh number mein hota haijo do dafa multiply karne sey aatahai”* (it is present in number which is obtained by multiplying a number twice).

I expected them to tell me that the square shapes is the multiple of length of its *two equal* sides and the arithmetic square is refers to a number being multiplied *twice*. The above scenario struck me because I was not prepared for these responses. Pedagogical content knowledge is very important for teachers to possess because it gives information about students thinking on specific areas and it prepares the teacher to predict student errors (Lau & Yuen, 2012; Peng, n.d). Shulman (1986) argues that through pedagogical knowledge teachers can identify students' misconceptions. I was not aware that the students will not be able to make the connections between the different branches of mathematics. Bosse (2003) says that connections in mathematics help students to use and remember a larger number of thoughts. Students can merge those thoughts in to connected mental componentsfor easy handling later.

Previously in my teaching I was not an advocate of connections within mathematics although I was aware of them. Maybe, I never focused on them because I did not know about the benefits of making connection. Now since I know about the benefits of making connections with mathematics I also want my students to learn that.

## TEACHERS KNOWLEDGE, THINKING AND INSTRUCTION

### Prime Factorization

#### Teaching

The topic of prime decomposition was to be taught by Jaweria and I was sitting at the back of class as an observer. The definition of prime factorization that she used was “when we decompose a composite number into prime numbers”. This definition showed that she had mathematical knowledge. However, the children’s faces were blank as they had not been able to grasp the definition because of complex language involved in it.

To me the definition appeared difficult although it was good one. As mentioned earlier the New Syllabus book of Oxford publishers was being used in the school which I think was a little inappropriate for students. By inappropriate I mean two things one it was difficult for their age and since the students were from middle and lower middle SES backgrounds the book needed more translation to become comprehensible and more relevant for them. A study by Macintyre and Hamilton (2010)shows that to maximize students’ engagement with mathematics, content selection of text books should be realistic and relevant to their experiences. In the present context realism and relevance to the students’ experiences was missing. As, the students’ primary language of communication was Urdu and hence a word like ‘decomposition’ was difficult for them to comprehend. Here I would suggest based on my experience that perhaps a more contextually relevant textbook be selected or the content of the given book be translated to match the students’ level.

**[...]**

^{1} Socio Economic Status

^{2} Urdu word for long

- Quote paper
- Asma Mamsa (Author), 2014, Least Common Multiples and Squares. Reflections on co teaching Mathematics in a community based school, Munich, GRIN Verlag, https://www.grin.com/document/1001524

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