Task 3: (Number sense)
The purpose of this essay is to outline the “Big ideas” that are crucial to a well-developed sense of numeracy in which will lead to the basis of explaining the role of number sense in everyday problem-solving and reasoning. The ‘Big ideas” include enabling people to recite, insert and evaluate digital numbers in everyday concepts, through a deep understanding of digital numbers and operations which is used throughout the curriculum to solve problems and reasons mathematically.
According to Luneta et al (2013:179), to understand a number sense it is crucial to firstly develop a direction of placing together the sense of measurement, data handling, space and shape, in which leads to the development of becoming a critical component to numeracy, whereby leads to the developed understanding of the ‘Big ideas’ behind it all. This includes the aspect of a number sense in which allows people and learners in schools to read over, include and assess numbers in everyday life circumstances, in which needs a deep understanding of numbers and operations. This leading to procedural fluency which is also necessary but is not always an adequate requirement for number sense as a number sense is used throughout the curriculum policy statement to solve brought up problems and reasons in a mathematical way.
Therefore according to Luneta et al (2013:186), a well-developed sense of number requires a profound and adaptable comprehension of numeration. This implies an informed knowledge basis of various number sets and how they are composed, spoken to, and related, together with a comprehension of the manners by which the tasks on number may be spoken to in various settings. For instance, according to Luneta et al (2013:186), addition is frequently comprehended as hostility (meaning getting more) and subtraction as (losing a few or all). For example, in the event of being R700 in debt, depositing an amount of R300 will bring about less obligation, however no more capital (e.g. –R700 + R300 = -R400). While this applies to addition and subtraction including positive whole numbers, it does not sum up to every number set. For example in figure 1 below, placing together the aspects of A and B could be seen as the outcome of 3 counters and 5 counters equalling a total of 8 counters (hostility), or it could be seen as the outcome of counters that are red prior to placing them together, one-third in the case of aspect A and one – eighth in the case of aspect B. When placed together two – elevenths are red. In this sense, it ).
Figure 1: Different explanations based on addition through the use of fractions
Aspect A Aspect B
Scaffolding on the other hand according to Luneta et al (2013:187) is a sense of direction that causes learner-learning to take place in the form of being a primary or smaller task for teachers involved in teaching mathematics as a subject. This is of great importance in connection with a relatively lower or smaller number of ‘big ideas’ and strategies in number, without which students results or outcomes in mathematics will suffer.
These “big ideas” according to Luneta (2013:187) are key to a well-developed sense of numeracy, in which are broken down and elaborated in various chapters in which are discussed based on the following:
Trusting the count
According to Luneta et al (2013:187) trusting the count is a valuable term which is based on the aspect of outlining that children do not always believe that if they had to count the same collective batch again they would receive the exact amount or number as before. Therefore, according to Luneta et al (2013:187) this term is placed at use in the form of being incorporated not only to outline the initial and literal interpretations but to also identify the capacity of a child’s access to flexibility of mental objects for the digital numbers of 0 to 10, whereby learners can speak about the number-naming patterns to 30 and above; seeing, reviewing and writing number digit words and numerals to 10; and to create the aspects of being able to count and model small collective operations from lower than 20, but will end up taking a guess when given the question of ‘how many’ in many particular aspects of which could include two-digit numbers in the sense of being written or verbally given as either being larger/smaller. Learners might experience great difficulties in the areas of counting bigger digit numbers (50 and above) accurately. According to Luneta et al (2013:187) this could be due to:
Not knowing that counting is a form of technique strategy that places emphasis on determining the value of ‘how many’.
Not being able to arrange the objects in the correct way due to already being counted.
Not having a great understanding based on the countable facts of 0 to 10.
Therefore, according to Luneta et al (2013:187) at the end of a learners first year in school, they need to develop a sense of direction towards understanding in great depth the number digits of 10, in which they see as represented and how they are configured in connection to other digital numbers. For instance, if learners see, write or hear the digit 9 they imagine the digit number to look and relate to a number of factors in which do not correspond necessarily to addition and subtraction but is seen to be broken down into aspects like 1 less than 10 =9 or 1 more than 8 = 9, as 4 and 5 = 9, or 3 and 6 = 9. This way it outlines the aspects of which leads to a deep understanding of what exactly each number means and the different ways it could be broken into to be represented as the digit 9 (Luneta et al, 2013:188). In which contributes to the development of a sense of number.
According to Luneta et al (2013:188) place value is a difficult term to outline in the form of either teaching or learning it, due to it being a hidden aspect behind the performance of counting. Therefore, many students are brought up with the ability to identify a place value in the form of breaking the parts of the place value digit into separate parts, for example outlining that there are 6 hundreds, 4 tens and 8 ones in the digit number of 648 in which leads to the ability of being able to count verbally to 100 and above. According to Luneta et al (2013:188) this could be due to learners:
not having sufficient knowledge of numbers being connected in which they do not have the belief of trusting their count;
not having the ability to see numbers such as 2 and 10 as countable terms;
no or little sense about number senses beyond the number count of the number digit of 10;
Failing to see the outcome basis of a structured platform in the sense of recording digital numbers such as 2-digit numbers.
Learners therefore have to develop a deep understanding of place value sequences or patterns by the end of their third year in school, in the form of it supporting the ways of working out or operating with 2 digit numbers and above.
According to Luneta et al (2013:188) place value parts are an unfortunate part in some cases due to it often being introduced to the learner’s way before they have to demonstrate an understanding that digit numbers such as 2 to 10 can be implemented in the form of being used as countable element units. Consequently, learners develop a misunderstanding that threatens their total capacity to form an understanding that has meaning, useful strategies for mathematical mental calculation and their later developed understanding of greater integers and fractions as a whole in which contributes to the development of a sense of number.
According to Luneta (2013: 189) thinking in the sense of multiplication enhances the thoughts behind uplifting the aspects of addition and subtraction. Whereby it is not the stated facts for multiplication purposes, although it does fall into sharing in form of gaining aspects together in which is used as a way to support and uplift a learner’s interaction with the created environment around them.
Therefore, according to Luneta et al (2013:189) learners should develop a direction of understanding multiplication aspects of between 50 and 100 on the platform of simple to adequate multiplication and division procedurals in the correct manner understood. Although most learners rely on “rote” learning in the form of working in groups with others by understanding the basis of multiplication by repeated aspects of addition. With as little or no possible access to a wider outcome of thoughts behind multiplication learners develop the aspects of finding it challenging to see forth in developing mental capabilities with the form of implemented strategies, and as a consequence to rely upon memorised patterns for multiplying and dividing in the sense of larger whole digit numbers and decimals. According to Luneta et al (2013:189) this is due to:
Not having the ability to place trust on the count of the number outcome and seeing the numbers as countable elements in their own unit space that can be viewed a number of times
Having a poor or low mental capacity based on developing strategies for the purposes of addition and subtraction.
Having no or as little as possible access to alternative model patterns of multiplication aspects of mathematics.
Therefore, according to Luneta et al (2013:189) learners should by the end of their grade 4 year have the ability to think about the aspects that make up multiplication in a number of ways to develop an understanding of how it links to division in which uplifts the written and verbal computation behind solving a wider range of problems faced in the form of including equal proportions, combinations and uplifted rates. In this way learners need to develop an understanding based on the advantages around and of representation as a whole in elementary terms such as arrays and regions which highlight multiplicative situations that help with problem-solving which leads to deeply understanding the reasoning behind each one in which contributes to the development of a sense of number.
An idea of collection and quantity behind understanding the sense of number is however identified as an essential and fundamental aspect that identifies its parts efficiently. However, according to Luneta et al (2013:191) learners tend to experience great difficulty with sizable proportions such as fractions, decimals and percentages. Therefore, learners also tend to misinterpret the great meaning behind the aspect of a denominator in the sense of working out a problem faced by a fraction in which they exhibit a proportional understanding based on the idea around it in which is limited to the extent that it forms into the context of a proper fraction. For instance, working out how many pink jelly totes are in a packet, whereby learners according to Luneta (2013:192) at this stage outline fractions to be numbers that have arrived on the basis of partitive division. For example, sharing 6 boxes of pizzas among 7 people. This could be due to:
Seeing the denominator as the numerator as well;
Limiting exposure to experiences to how numbers parts are named accordingly;
There only being a grouped up idea for multiplying and dividing; and
Hardly any or no access to strategies that could build up appropriate cases of fraction representation.
Therefore, according to Luneta et al (2013: 192) learners should develop the aspects of working with understanding the meaning behind a variety of numbers. In which they need to establish a meaningful basis of thinkable patterns based on rational digit numbers such as mixed fractions, proper fractions, percentages and decimal fractions. This including the aspects behind involving equal measures, increasing and decreasing where numbers become either smaller or bigger and fraction representation for the equal parts that develop over the use of quantities.
Understanding the connection between fractional parts and division in the sense of partitive division is an essential aspect in outlining the fractional renaming of equivalence whereby learners according to Luneta et al (2013:193) need to understand the continuous ideas behind multiplication in the form of linking it to fraction diagrams in which contributes to the development of a sense of number.
The reason behind gaining proportional reasoning is that due to many finding it difficult to interpret various sorts of using ratio, rates and percentages it is due to not having the correct mental capacity to use it in its correct sense. Therefore, according to Luneta et al (2013:192) proportional reasoning is formed on the basis of depending and relying upon many interconnected strategies with leading ideas that take time to develop. At the centre of its element this sort of reasoning requires the capacity of identifying the aspects needed to be compared with the following of a suitable description to what it’s precisely being compared to. This forming the aspect of two reasoning patterns behind certain proportions which include both needing some sort of comparison technique to take place. Therefore it is easier to identify what is being compared in the context of quantities used, however they are represented in which involves all the variables at use.
Luneta et al (2013:192) also states that proportional reasoning involves the work behind the capacity of flexibility and confidently with the quantities included in the sense of gaining the ability of seeing multiplicative connections in the variety of its problems faced in the context of involving rational digit numbers and their operators in which would contributes to the development of a sense of number.
To gain the depth understanding behind generalisation according to Luneta et al (2013: 194) would be that it could be considered as the use of numbers in earlier levels of childhood, whereby many learners are able to operate with the use of rational digit numbers to the extent that they would have to emerge on appreciation of the real digit numbers. However, according to Luneta et al (2013:194) also stating that it is not generally the case when digit numbers are presented as underlying pronumerals or in use of expressing pronumerals.
Therefore, according to Luneta et al (2013:194) building algebraic texts helps to describe and explain the connection in which is also seen to be a difficult areas whereby learners find it difficult to create deep understanding. A variety of external presentations are then used to outline connections and strategies to form greater understanding based on the problems faced today with problems that need strategies and techniques to solve mathematically. Therefore it is seen to be at a great level of difficulty for most in which may be caused by:
Differentiation of interpretations ;
Limiting understanding of number operates;
Little or no access to communicating between connectional mathematical concepts; and
Only developing a sense of direction based on multiplicative and proportional reasoning.
Therefore, according to Luneta et al (2013:195) learners should develop a working strategy based on developing a meaningful basis of a variety of numbers including the mathematical connection between equations, identities, relations and functions in which could lead to understanding numbers and their operators in an efficient manner that contributes to the development of a sense of number.
Problem-solving and reasoning
Problem-solving and reasoning on the other hand include the ‘big ideas’ discussed above in which are discovered in our daily lives whereby they play a crucial role in developing a number sense throughout each and every day that one lives. Problem-solving and reasoning are important factors when it comes to facing problems such as mathematical problems throughout life itself in which allow people to breakdown the problems faced into smaller parts in which can be addressed with finding suitable solutions to apply reasoning factors behind certain reasoned problem solving methods to gain deeper understanding about topics based in mathematics. Therefore, people need to gain the insight of applying the correct measures of problem-solving methods and reasoning such as:
Trusting the aspects of one’s own counting when needed, for instance counting money or stock in shops, whereby it is crucial to develop the sense of trusting the result the first time around in order to save and spare time where needed in the form of applying suitable problem-solving methods with developed reasoning to ensure it is also correctly done the first time around. This could also be addressed by additive and subtractive reasoning in which help develop the instances of variables in which are needed to be used in cases of finding the most accurate solution mathematically.
One also needs to apply the correct place value aspects in the form of when problems arise in order to solve them correctly in the measures of accuracy with the use of any sort of numbering system that requires problem-solving methods in everyday situations whereby it is crucial to gain the understanding and of each method used mathematically with the reasons behind that specific problem, leading to the outcome of finding a suitable reasoning aspect to solve the problem faced at the time during the day.
Another idea behind problem solving and reasoning would be thinking multiplicatively in the sense of numbers being repeated in which may cause problems that would needed to be solved in the sense of applying a suitable reason behind the problem to come up with a solution to solve the problem mathematically in the sense of using numbers adequately and continuously to find the best result to address the problem mathematically. For instance, problems such as developing to many correspondence in a mathematical problem would lead to a multiplicative form of reasoning in which would address the problem in various ways with the use of various mathematical methods accurately in the sense of using numbers efficiently to develop the correct reasoning aspects behind that specific problem faced during our daily lives.
Partitioning on the other hand may develop the problems behind proportional sizes based on decimals, fractions and percentages whereby it may be uplifted in the sense of erupted problems in which problem solving and reasoning would have to take place in the sense of guiding a solution or reason to fix the problem in which might be that a percentage of the profit made in a business is wrong at the end of the day in which would have to be recalculated using formulas or mathematical methods based on partitioning to fix the problem with a suitable reason behind it with the use of a deep understanding of using the correct methods of breaking the problems faced into parts to address them separately.
Proportional reasoning then falls in the place of which may also lead to problems in which need to be solved or given a suitable reason behind why it may have happened in the sense of it calculating rates at the wrong amounts or ratios of interpretation whereby problem solving techniques or reasoning would have to be implemented in the case of fixing the proportion to develop the most accurate solution for the problems that may be faced using appropriate mathematical methods with the deep understanding gained behind knowing what may be included behind the reason of proportional reasoning and its workings within our daily lives.
Generalising as another sort of big idea behind the development of a number sense could also implement the aspects of using problem solving and reasoning to find suitable solutions in fixing the problems that arise in everyday situations like building up wrongful connections between inaccurate circumstances in which would apply to finding a suitable and well developed reason to outline a needed solution to address the problem accurately with the use of gained knowledge based on the formation of mathematical methods taught to apply a sufficient amount of understanding when addressing the problems faced when generalised in our daily lives.
Therefore, it is important to implement the thoughts behind various problem-solving techniques and reasoning to cover up and correct the problem that may be faced or may already have been faced but a problem solving technique or reason needs to be put in place to address the problem adequately in order to engage in the position of contributing to the sense of numeracy overall in a mathematical fashion. That is why it is important to develop deep understanding when taught about the ‘big ideas’ behind a number sense.
In conclusion the purpose of this essay was to outline the “Big ideas” that are crucial to a well-developed sense of numeracy in which will led to the basis of explaining the role of number sense in everyday problem-solving and reasoning. The ‘Big ideas” included enabling people to recite, insert and evaluate digital numbers in everyday concepts, through a deep understanding of digital numbers and operations which are used throughout the curriculum to solve problems and reasons mathematically.
Luneta, K., 2013. Teaching Mathematics: Foundation and Intermediate Phase. First ed. Cape Town: Oxford University Press Southern Africa.