Teaching Maths Through Culture

Teaching Maths Through Culture Culture could be defined as the totality of a person’s way of life. This has to do with everything in an environment including environmental problems and activities. Mathematics on the other hand could be the way we think about things around us or science of pattern and order. For the mathematical instruction to improve the achievement and interest of the learners there is need for mathematics teaching that has the learners’ cultural background. Culture as a tool for conveying mathematical ideas is the experiences of all aspects of mathematics as it is found in a human tradition and culture. Every society has an intuitive kind of mathematical knowledge due to the importance of mathematics in terms of their way of counting, measuring, relating, classifying and inferring. It is surprising that much of this knowledge have been ignored in the formal school mathematics curriculum as most of our texts use foreign illustrations that are not so common within the locality. There is a great need to connect mathematical content and the home cultures of learners as well as between different branches of mathematics at this early stage. Most parts of the world are exploring and acknowledging the need for connecting

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mathematical content with the real world of work. Teachers’ cultures to a certain degree do have influnce on teachers’ perception towards their pupils’ attitudes and academic performance in schools. The challenge for mathematics teacher educators often has more to do with how to help teachers “unlearn” the wrong ideas and help them learn the correct concepts. For example, in the Philippines, the problem of mathematics teachers' continuing education is compounded by the reality that many of them do not know enough content and possess limited pedagogical content knowledge (Catherine P. Vistro-Yu 2007)

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Knowing mathematics does not only mean learning mathematical concepts and processes but also knowing and understanding the cultural practices that go with the learning of these concepts and processes. Thus, mathematics is a cultural product. The mathematics that is learned in school is not the same in many aspects because of the unique cultural traditions, language, practices, and meanings that have been developed and passed on by members of the group that work together on it. Mathematicians have long developed their own tradition and culture with regard to the ways of knowing, learning and talking about mathematics. It is sometimes difficult for students to appreciate the importance of Mathematics. They often find the subject boring and hard to understand. By sharing teachers’ culture and traditions with pupils’ cultures, hopefully it would help pupils realise that Mathematics is not just a subject on their time-table but a tool they use in their everyday life and their way of life Much of the intuitive knowledge gathered from the society in terms of counting, measuring, relating and classifying has been ignored in the formal school mathematics curriculum. The need to connect mathematical content and the home culture of learners is paramount for enhanced performance in mathematics at the primary school level. Culturally based materials in this context could be those local materials found within the environment of the learner that could be used to convey mathematics concepts in its real form. For example, (i) Geometric concepts: ‘ketupat’ (ii) Folk songs, Riddles and Puzzles Folk songs like ‘two’, four’, ‘six’ or ‘five’, ‘ten’, ’fifteen’ are also used in teaching mathematical concepts. This song involves two or more people forming a circle and deciding on whether to count in ‘twos’ or ‘fives’ and the maximum number they will count. As people count, any number that does not fall in, they are dropped until a winner emerges. This game is used to teach the concept of multiples, sequence and base number system in mathematics.

(iii) Telling Time 2

‘cock crow’. The cock crow in early hours of the morning signifies morning period{4am to 6am). cockcrow at dawn. Also, in the evening period towards 6pm, the cockcrow(dusk) shows that the day is getting to an end. Therefore, the concept of time can be taught through this cultural means by relating time with those familiar terms used for different period of the day. Shadow. Whenever the shadow is getting longer in the morning in the front, it means the day is just starting (school time). As the day goes down, the shadow enlongates. At 12 noon the shadow gets longer again and goes behind you. Towards evening, the shadow gets to the wall. The concept of time is also communicated through this way. (iv) Cylinder: Bamboo “pansuh” Activity 1 Choose a culture or tradition of your pupils or your own culture and show how to use it to teach mathematics in schools.

Attaining Concepts Model Concept attainment is “the search for and listing of attributes that can be used to distinguish exemplars from non exemplars of various 3

categories.” Concept attainment requires a pupil to figure out the attributes of a category that is already formed in another person’s mind by comparing and contrasting examples (called exemplars) that contain the characteristics (called attributes) of the concept with examples that do not contain those attributes. Whereas concept formation, which requires pupils to decide the basis on which they will build categories. To create such lessons we need to have our category clearly in mind. Examplars Essentially the exemplars are a subset of a collection of data or a data set. The category is the subset or collection of samples that share one or more characteristics that are missing in the others. It is by comparing the positive exemplars and contrasting them with the negative ones that the concept or category is learned. Attributes All items of data have features, and we refer to these as attributes. For example, nations have areas with agreed on boundaries, people, and governments that can deal with other nations. Cities have boundaries, people, and governments also, but they cannot independently deal with other countries. Distinguishing nations from cities depends on locating the attribute of international relations. To teach a concept, we have to be very clear about its defining attributes and about whether attribute values are considered. We must also select our negative exemplars so that items with some but not all the attributes can be ruled out. There are two ways that we can obtain information about the way our pupils can attain concepts. After a concept has been attained, we can ask them to recount their thinking as the exercise proceeded- by describing the ideas they came up with at each step, what attributes they were concentrating on, and what modifications they had to make. (“Tell us what you thought at the beginning, why you thought so, and what changes you had to make.”) This can lead to a discussion in which the pupils can discover one another’s strategies. Many people, on first encountering the concept attainment model, ask about the function of the negative exemplars. They wonder why we should not simply provide the positive ones. Negative exemplars are very important because they help the pupils identify the boundaries of the concept.

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For example, if we select numbers that are odd numbers (these become the positive exemplars) and some that are not (these become negative exemplars- the ones that do not have the attributes of the category odd numbers). We present the numbers to the pupils in pairs. Consider the following four pairs of numbers: Example 1 121 81 11 5

187.1 26.2 1 2 8

To carry on the model, we need about 20 pairs in all- we would need more if the concept were more complex than our current example, that is, odd numbers. We ask the pupils to make notes about what they believe the exemplars have in common. Then we present more sets of exemplars and ask them whether they still have the same idea. If not, we ask what they now think. We continue to present exemplars until most of the pupils have an idea they think will withstand scrutiny. At that point we ask one of the pupils to share his or her idea and how he or she arrived at it. If some pupils still cannot create the concept from the given exemplars, teachers should provide some more examples until the pupils agree that each positive exemplar adds something to the meaning of the type of numbers. We continue by providing some more numbers and by asking the pupils to identify the numbers that belong to our concept (odd numbers) and ask them to agree on a definition. The final activity is to ask the pupils to describe their thinking as they arrived at the concepts and to share how they used the information that was given. For homework we ask the pupils to find odd numbers in the list of numbers given to them and the application of odd numbers in daily life. We should examine all the exemplars they come up with to check that they have come up with the correct concept. Example 3 5

Present the following list of numbers labeled yes or no 21 22 42 32 27 17 33

yes no yes no yes no yes

TEACHER: I have a list of numbers here. Notice that some have “yes” by them and some have “no” by them. (Children observe and comment. Teacher puts the list aside for a moment.) Now, I have an idea in my head, and I want you to try to guess what I’m thinking of. Remember the list I showed you. (Pick up the list.) This will help you guess my idea because each of these is a clue. The clues work this way. If a number has a ‘yes’ by it (points to first number), then it is an example of what I am thinking. If it has a ‘no’ by it, then it is not an example. (The teacher continues to work with the pupils so that they understand the procedures of the lesson and then turns over the task of working out the concept to them.) TEACHER: Can you come up with the name of my idea? Do you know what my idea is? (The pupils decide what they think the teacher’s idea is.)

TEACHER: Let’s see if your idea is correct by testing it. I’ll give you some examples, and you tell me if they are ‘yes’ or ‘no’, based on your idea. (The teacher gives more examples. This time the pupils supply the ‘nos’ and ‘yeses’.) 30 33 10 9 4

(yes) (no) (yes) (no) (yes)

TEACHER: Well you seem to have it. Now think up some numbers you believe are ‘yeses’. The rest of us will tell you whether your example is right. You tell us if we guessed correctly. (The exercise ends with the pupils generating their own examples and telling how they arrived at the concept.)

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In this lesson the pupils should be able to identify the concept, that the numbers are not divisible by 2 as odd numbers. Example 4 20, 35, 46, 55, 12,

15, 4 (No) 12, 6 (No) 40, 6 (Yes) 36, 19 (Yes) 7, 6 (No)

In this example, children should be able to identify that the operation involves is subtraction. That is the third number is obtained when you subtract the second number from the first number. According to research findings, concept teaching provides a chance to analyze the pupils’ thinking processes and able to help them developing more effective strategies.

Activity 3 In a group of 3-4 persons, plan an activity using concept attainment model to teach: 3.1 3.2 3.3 3.4 3.5 3.6

multiplication. place value. decimals. fractions. time. percents. 7

3.7 measurement: length. 3.8 measurement: weight.

Problem Posing The focus of this section is to generate new problems and questions base on the existing given problems. There is always a fixed number of ways (combinations) in which questions can be asked. You must find all the different combinations of questions for each chapter to be tested and find out the steps needed to solve it. Once you have learnt the steps to solve each type of question, the next step is to internalize it by practicing doing each type of question. Steps of Writing All the Possible Ways to Ask Questions Step1: Collect different types of combinations from a question 8

Collect all the possible different types (combinations) of questions for each chapter. You search for all these possible questions from a variety of sources. For example, a compilation of examination questions over the last ten years, assessment books, homework from school, past test papers and the test papers of other schools (especially the elite ones). Step 2: Learn the steps to solve For each of the question types you have collected, find out the steps needed to solve each one. You will find that for a particular type of question, the steps involved are the same, even though the numerical numbers may vary. Step 3: Practice to internalize Finally, you must practice doing each type of question at least three times, using the steps needed to solve each one. There are many pupils who diligently go through hundreds of question types and practice solving them, yet still run into new question types that stump them during the exam. This is because they just practice on questions at random.

For example, in a chapter (let’s call this chapter X), you will find that there are a fixed number (N) of possible question types. They are X1, X2, X3, X4, ……Xn. Every question type requires a different set of steps or skills to solve. However, you will find that for each type of question (lets say X1), there are many possible variations that could come out for the exam. This will be X1a, X1b, X1c ….etc. Many variations of a particular type of question are generated by changing the numerical values involved. They might be an infinite number of variations that can be used to ask a question for each type, but all the different variations of the same type of question can be solved using the same formula or steps. For example, if you can solve one (that is, X1a) , you should be able solve the rest (X1b, X1c, X1d…..etc).

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Activity 4.1 In a group of 3-4 persons, write down all the steps showing how you get the answer to the following question. Base on your answer, list down all the variables in the question. Generate all the possible questions using each of the listed variables. The diagram below consist a triangle and a square. If the perimeter of triangle PQR is 19 cm, find the perimeter, in cm, of the whole diagram.

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Solution

Variables

Generated Questions Using the Listed Variables

Activity 4.2 In a group of 3-4 persons, write down all the steps showing how you get the answer to the following question. Base on your answer, list down all the variables in the question. Generate all the possible questions using each of the listed variables. Rentap jogs 2.65 km. Ah Meng jogs 600 m more than Rentap. What is the distance covered by Ah Meng, in km? Solution 11

Variables

Generated Questions Using the Listed Variables

Activity 4.3 In a group of 3-4 persons, write down all the steps showing how you get the answer to the following question. Base on your answer, list down all the variables in the question. Generate all the possible questions using each of the listed variables.

Mr. Ling spent RM245.90 in week. Mr. Abu spent twice of the expense of Mr. Ling. How much money did Mr. Abu spend?

Solution 12

Variables

Generated Questions Using the Listed Variables

Activity 4.4 In a group of 3-4 persons, write down all the steps showing how you get the answer to the following question. Base on your answer, list down all the variables in the question. Generate all the possible questions using each of the listed variables. A glass contains 360 ml of water. How many of these glasses of water are needed to fill up a jug with a capacity 1.8 liter? Solution

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Variables

Generated Questions Using the Listed Variables

Activity 4.5 In a group of 3-4 persons, write down all the steps showing how you get the answer to the following question. Base on your answer, list down all the variables in the question. Generate all the possible questions using each of the listed variables. The population of a village is 69,482. During the flood 3,195 people move out of the village. A few days later, 2,937 people move back to the village. Calculate the population of the village now. Solution

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Variables

Generated Questions Using the Listed Variables

Activity 4.6 In a group of 3-4 persons, write down all the steps showing how you get the answer to the following question. Base on your answer, list down all the variables in the question. Generate all the possible questions using each of the listed variables. A total of RM12450 is raised in a charity event. RM2680.50 is donated to fire victims. The balance is donated equally to 3 orphanages. How much does each orphanage receive? Solution

Variables

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Generated Questions Using the Listed Variables

Activity 4.7 In a group of 3-4 persons, write down all the steps showing how you get the answer to the following question. Base on your answer, list down all the variables in the question. Generate all the possible questions using each of the listed variables. A printing machine can print 1500 copies in one hour. If the number of copies is reduced by 10%, how many copies can it in 2 hours 40 minutes? Solution

Variables

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Generated Questions Using the Listed Variables

Activity 4.8 In a group of 3-4 persons, write down all the steps showing how you get the answer to the following question. Base on your answer, list down all the variables in the question. Generate all the possible questions using each of the listed variables. Jenny makes 6 pairs of trousers for children. After making the trousers, she had 2 m 76 cm of cloth left. If she had 12 m of cloth at first, how long of cloth did she use for each pair of trousers? Give your answer in m and cm. Solution

Variables

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Generated Questions Using the Listed Variables

References Adam Khoo, (2006). I am Gifted, So Are You. Marshall Cavendish Edition. Singapore. Catherine P. Vistro-Yu (2007) Enhancing Mathematics Teachers’ Professional Development Through Shared Cultures Ateneo de Manila University, Philippines: EARCOME4 Joyce, B. et al (1992). Models of teaching. Allyn and Bacon. Massachusetts

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