The understanding of integers and negative numbers in particular poses certain challenges for many people. One difficulty lies with the fact that the negative sign appears along with a number to indicate an object, a number, but the same... more
The understanding of integers and negative numbers in particular poses certain challenges for many people. One difficulty lies with the fact that the negative sign appears along with a number to indicate an object, a number, but the same negative sign is also used for the operation of subtraction
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Concealed behind children’s daily activities, there is a lot of mathematics comprising counting, comparing, estimating, recognising patterns, sequencing, logical thinking, and reasoning. By including all these aspects, the Sikkim SCERT... more
Concealed behind children’s daily activities, there is a lot of mathematics comprising counting, comparing, estimating, recognising patterns, sequencing, logical thinking, and reasoning. By including all these aspects, the Sikkim SCERT Math books make the subject more child-centred, less abstract, and more appealing
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Practically every human activity involves usage of thinking skills. What are thinking skills? They are essentially mental processes that we do: classifying objects, observing properties, encoding information, comparing, taking decisions,... more
Practically every human activity involves usage of thinking skills. What are thinking skills? They are essentially mental processes that we do: classifying objects, observing properties, encoding information, comparing, taking decisions, making inferences and solving problems. Thinking skills can be viewed as the building blocks of the whole canvas of thinking. These thinking skills are broadly classified into two categories: lower order thinking skills and higher order thinking skills
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Measurement occupies a unique position in the curriculum, for various reasons. As it is an essential everyday activity in human life, children are naturally exposed to measurement in various situations at home and elsewhere. Also,... more
Measurement occupies a unique position in the curriculum, for various reasons. As it is an essential everyday activity in human life, children are naturally exposed to measurement in various situations at home and elsewhere. Also, measurement overlaps both with numbers and geometry. It involves spatial dimensions as well as counting. In measurement, one is measuring one attribute in terms of another attribute. Also one is expressing a non-discrete quantity in terms of discrete numbers. Since there are different ways of measuring the given length or mass, the choice of the measure is dependent on the purpose that needs to be served. Children need to understand that some contexts require precision to a fine degree while some require approximate figures. A strong foundation in measurement concepts leads to a better understanding of decimal numbers in particular
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Teaching of the place value system happens in the context of teaching numbers and is very closely related to counting, grouping objects to aid counting, usage of number decomposition, learning the patterns in number names, learning the... more
Teaching of the place value system happens in the context of teaching numbers and is very closely related to counting, grouping objects to aid counting, usage of number decomposition, learning the patterns in number names, learning the written representations of numbers, learning the patterns in the relationships between consecutive places, and developing a proper number sense. Children develop facility with numbers and a sound understanding of the number system only if sufficient care is taken in building all the above mentioned areas
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In the article on ‘Recurring Decimals’ published in the November 2013 issue of At Right Angles, many questions remained unanswered in the end. They had emerged as empirical observations during the course of the exploration. We study these... more
In the article on ‘Recurring Decimals’ published in the November 2013 issue of At Right Angles, many questions remained unanswered in the end. They had emerged as empirical observations during the course of the exploration. We study these observations closely here, examine their validity and explain them using simple principles of divisibility
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How does one introduce a topic like ratio, which is so widely present in daily life and so intimately connected with human experiences? Our cherished cultural achievements are permeated with it: music is full of ratios, as is art. Our... more
How does one introduce a topic like ratio, which is so widely present in daily life and so intimately connected with human experiences? Our cherished cultural achievements are permeated with it: music is full of ratios, as is art. Our daily existence involves cooking and shopping, and these are filled through and through with the usage of ratio. Shadows, which are present with us all through the day, offer a visual depiction of ratios in action. In mathematics as a subject, the notion of ratio is embedded into many topics – sometimes in an obvious way, at other times not so. Fractions, scale drawing, enlargement, trigonometry, tables, linear equations… are all illustrations of this. Considering that ratio is linked in an essential way to so many concepts in mathematics, it is important to lay a strong foundation and develop the necessary conceptual base when teaching this topic. A basic error that often arises while studying ratio is the application of additive thinking to a context...
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Most assessments conducted across the country indicate that the first stumbling block for many children is the subtraction operation, followed by division. On looking more closely we see that the difficulties often arise in subtraction... more
Most assessments conducted across the country indicate that the first stumbling block for many children is the subtraction operation, followed by division. On looking more closely we see that the difficulties often arise in subtraction contexts involving double digit or larger numbers. This difficulty is largely caused by three factors: (i) improper understanding of place values (ii) lack of understanding of the rationale behind the formal subtraction procedure, (iii) not seeing the connection between addition facts and subtraction fact
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Area and perimeter are forms of measurement that are used commonly in many day-to-day activities. In particular, area is used in an intuitive way on an everyday basis when we select a plate to cover a utensil, a table cloth for a specific... more
Area and perimeter are forms of measurement that are used commonly in many day-to-day activities. In particular, area is used in an intuitive way on an everyday basis when we select a plate to cover a utensil, a table cloth for a specific table, a sheet of paper to cover a book, etc. Without really knowing the specific words, children also commonly make judgements which involve an intuitive understanding of area. A question that naturally arises is: When or why do we want to know the exact size of a space? Demonstration of this point needs to happen repeatedly through real world applications
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We first expose children to the topic of converting fractions to decimal numbers in class 5 or 6. At that point children notice that some fractions terminate and some do not, and they come across terms like terminating decimals and... more
We first expose children to the topic of converting fractions to decimal numbers in class 5 or 6. At that point children notice that some fractions terminate and some do not, and they come across terms like terminating decimals and recurring decimals. They are also shown the usage of the bar or dot notation. Generally most textbooks do not proceed beyond this point. Later (class 8 or 9) they are taught how to rationalize numbers. The activity I describe here is one which I have tried with class 8 children. It proved to be an interesting investigation into the patterns in recurring decimals leading to generalization and looking at the reverse process initially through a trial and error approach followed by arriving at the procedure for rationalization
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Introduction to school algebra can happen through varied approaches. Some prefer to start with an unknown in an equation, while some prefer to start with a formula and some others may prefer to use a pattern based approach. Does it make a... more
Introduction to school algebra can happen through varied approaches. Some prefer to start with an unknown in an equation, while some prefer to start with a formula and some others may prefer to use a pattern based approach. Does it make a difference which approach one uses? Is one approach better than the others? These questions can be debated. However, each of these approaches relates to different conceptions of algebra
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Time is one concept that is used by human beings on a daily basis, and perhaps in an instinctive way by animals and plant life too! The cock knows when to go ‘cock-a doodledoo’. Flowers know when to open their petals. Trees know when to... more
Time is one concept that is used by human beings on a daily basis, and perhaps in an instinctive way by animals and plant life too! The cock knows when to go ‘cock-a doodledoo’. Flowers know when to open their petals. Trees know when to shed their leaves. Children get exposure to the concept of time organically, well before they come to school. Yet the teaching and learning of time poses peculiar challenges. Time is an abstract concept. It involves measurement of something that is invisible and intangible. Consequently, challenges arise in understanding the concept of time. Learning the mechanics of reading time can also be a difficult task. It is therefore important to build the scaffolding carefully and match the activities to the children’s level of understanding
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The teaching of the geometry when approach in the right way holds an immense potential for learning the art of seeing and observing
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I currently teach coordinate geometry in Class 9 (ICSE Curriculum) and it is not clear to me why the basics of coordinate geometry are not included in the school syllabus (in India) in primary and upper primary levels. My experience of... more
I currently teach coordinate geometry in Class 9 (ICSE Curriculum) and it is not clear to me why the basics of coordinate geometry are not included in the school syllabus (in India) in primary and upper primary levels. My experience of using it with young students has been enjoyable. Coordinate geometry can be introduced at an early stage, in both the primary and upper primary levels
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Outline: The article is about connecting dots of parallel lines in a mathematical way. The experience of 30 years in teaching mathematics came up with examples & activities that show how students should develop a geometric eye
Introduction of a complex topic is always a challenge. If one intends to root it in the student’s daily experience, one is compelled to select a concrete model which serves the purpose but may have limited scope. At some point, the... more
Introduction of a complex topic is always a challenge. If one intends to root it in the student’s daily experience, one is compelled to select a concrete model which serves the purpose but may have limited scope. At some point, the student will need to abstract out the general notion in order to build a broader sense of the concept. The topic of ‘Equations’ can be approached in several ways. The choice of approach has a strong impact on the conceptual image which a student builds about a given concept. Hence, the choice is crucial in helping a student in understanding the concept as well as in developing the procedure for solving the problems. However, every approach has its limitations and can be used only for solving certain types of problems. Its use is limited and it may become necessary to expose students to other approaches when the type or complexity of the problems alters
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Measurement of line segments and the concept of length arise naturally in students. However, measurement of a turn or an angle formed between two rays is more complex to understand. Comprehending the concept of a degree as a measure of a... more
Measurement of line segments and the concept of length arise naturally in students. However, measurement of a turn or an angle formed between two rays is more complex to understand. Comprehending the concept of a degree as a measure of a turn and developing the skill of using a protractor takes time
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These activities comprise a possible approach that can be used at the Upper Primary level to revisit the concept of fraction, and deepen their understanding of the various rules used in the arithmetic of fractions