Note to Client:
Since it is a requirement that the mathematical knowledge is about primary school mathematics, the writer has changed the algebraic topic to solving fractions. Hope this meet your requirement. Thanks
Mathematical Knowledge Audit
Audit of Mathematical Knowledge
The primary schools teachers are required to have specific minimum qualifications in mathematics. In this regard, being a primary teacher, I must be able to learn different subjects in mathematics. And learning requires a lot of patients and adequate mathematical skills for the students to easily determine and cope with these subjects. To be able to determine whether a student is learning or not, subject knowledge assessment are being conducted or subject knowledge audit are being done. It is noted that to be a good primary school teacher, one must have substantial experience of the type of mathematical activities includes in the national curriculum mathematics which include practical work, problem-solving, and mathematics investigation. As a student, I am able to learn different mathematical subjects and knowledge relevant to primary school teaching courses. One of the lessons I have learned is about, addition of fractions. (Accordingly, the level of my subject knowledge can be categorised which corresponds to the need for my significant remedial support and others). Fractions consist of two numbers which is the numerator (top number) and denominator (bottom number). To add fractions, the denominator of two fractions must be the same. One of my strengths in adding fractions is in terms of finding the least common denominator of the fractions. For instance, in finding the sum of, the least common denominator is 6. By renaming the fractions using LCD we can get and the sum is . Another strength that I have in solving fractions is my ability to analyze, extend and create variety of patterns which could help me in understanding more the fraction problems given. For example, I have the skills in solving fractions with different denominators and I can also assist the students in doing this appropriately. Aside from these strengths, I also have the ability to learn formulas and apply them to the fractions which enables me to teach students the easiest way in solving the problems. Solving fractions requires the ability to think logically which solutions to be considered. In this regard, one of my weaknesses in fractions is when it comes to simplifying fractions especially improper fractions and also in finding the least common denominator (LCD) and the greatest common factors to find the LCD.
Other areas that I think would need further development for me as a student includes subtracting fractions, division of fractions improper fractions, Since the understanding and comprehension of fraction includes coordination of various mathematical contexts like rational number, division, equation and variable, this challenges me on how I will be able to solve these mathematical problems. I still need further development with these subjects for me to develop my understanding of rational numbers to help me in this mathematical subject.
It can be said that the extension of concepts from fractions to other aspects of fraction is a complicated process since it involves many sub-domains that I as a students have difficulty in comprehending like rational numbers, division, mixed fractions, simplifying fractions and others. The subject knowledge audit seems to have some impact on the ability of students as well as teachers to reflect on their own mathematical skills and thinking. Hence, knowing and understanding more mathematics or comprehending it more thoroughly seems to contribute to being able to think properly and effectively and in greater depth regarding what is consisted in doing mathematics. However, it can be said that there are people with higher subject qualifications who find it uneasy to determine their own thinking and skills and determine the steps which they take in solving, for instance, algebraic problems. Nonetheless, some students who had weaker subject knowledge show more ability to reflect in their own skills and develop a deeper knowledge of what is included in mathematical skills than what have been expected. Finally, it can be noted that subject knowledge on entry is of limited value in line with forecasting performance on the works of students. It can be said that students can development their skills and understanding to show a good level of competence by reflecting what their strengths and weaknesses not only of their mathematical skills but also the teaching and learning aspects.
Task 1
The relationship between children’s learning and thinking regarding mathematics and mathematics instruction are still not clear among scholars. The case in these shows that children’s inaccuracy of counting numbers is highly determined with how children construct their own knowledge and invent their own approaches in encountering numbers. According to Gelman & Gallistel (1978), the counting of children involves five principles which include one-to-one principle which involves marking off items in an array with one tick for each item, the stable-ordering principle in which number tags are repeated in stable order; the cardinal principle, where the last tag symbolize the number of items in the array; the fourth principle which permits any collection or array of items to be counted and the order-irrelevance principle that determines that the order of tagging is not relevant.
It has been indicated that children needs direct and complete representation to ensure accuracy on their mathematical knowledge. Herein, they develop their flexibility and abstractness and use counting to establish number facts to derive strategies. Accordingly, the representation of external objects as well as the manipulation of object or their representation and symbol is central to mathematics.
Theories which explain how children learn on counting are different with respect to development of counting processes. A child who comprehends the principles can be said to understand, consciously and unconsciously know the errors they make. It can be interpreted that children’s inaccuracy of counting is based on their behavioural principle and their use to form logical and numerical concepts.
In this regard, the role of the teacher in helping children have accurate counting is very important by correcting their mistakes and giving them more counting activities to enhance or improve their counting. The teacher should know how to interpret where the current condition of the child belongs in the five principles to know which remedies should be given. Children on one hand must be able to know that their counting abilities are prone to errors and they must accept these errors to easily learn some more mathematical skills in counting.
Task 2
There is a mathematical statement that all square is a rectangle. However, this statement can be confusing to children because children are taught that squares and rectangles have different definition. Squares are defined as a polygon with four equal or congruent sides while rectangle has been described to children as a polygon with fours sides and four right anglers. It has been noted that squares are a special type of rectangle where all four sides are equal in length and has four right angles
With this definition it follows that a rectangle has two pairs of parallel sides that makes it parallelogram. In geometry most children are taught that square and rectangle are two different shapes. In this regard, children are having difficulties in accepting that squares are rectangles because of their earlier conception of the definition of squares and rectangles.
Various studies also show that children are being confused in accepting square as rectangle because of their cognition on visual materials than theoretical ones. However, there are some children who can recognize square as a rectangle because these children are less predisposed or their prototype for rectangles possible distinguished from that of the square or these children are less able to judge the equality of all sides (Clements et al, 1999).
Task 3 A.
The focus on mental approaches and methods of calculation has given many students the important knowledge and recall skills necessary to undertake the routines embedded in formal written method. In the give cases of children, it can be said that each of the children are able to use the patterns they know in order to solve mathematical problems or arithmetic. Herein, it can be noticed that the Amy uses counting up from using the fingers and the result is correct. On one hand, the second case used different kind of mental calculation which he separate first 20 and add this to 67 and add 3 and later 3 to get 92. Note that both the first 2 cases have a correct answer while the third child has incorrect answer. This indicates that mental calculation among children may only be necessary for some cases and it has a tendency to provide wrong answers the correct ones. In this regard, the teacher must be able to let the children know which mental calculation is accurate at all times and which are not.
Task 3 B.
Multiplication
One of the mental calculation methods, for multiplication is the method for multiplying by 9. Herein, the skills needed are very easy because it only needs the knowledge of subtraction. Since, 9=10-1, multiplying by 9 is done by multiplying the first factor by 10 and then subtracting the original number from the result. For instance,
28 x 9 = 28 (10)-28= 252. This type of mental calculation method is only used if one of the factors is 9. This only needs your knowledge in multiplying by ten and subtracting.
In line with division, mental calculation includes the use of factors, or by splitting the number that you are dividing to make it simpler. For instance, 258 ÷ 6, the student may consider the factors of 6 (2 x 3) and can have 258 ÷ 2 ÷ 3 = 43. In this regard, the skills needed for the method to be effective is the ability of factoring,
Grid Method
Calculate 147 x 24 using (a) the grid method
Grid Method is a form of long multiplication utilising Partial Products Algorithm. First the multiplicand and multiplier are being partitioned into their place values then multiply where each table.
Total
x
100
40
7
20
2000
800
140
2940
4
400
160
28
588
Total
2400
960
168
3528
Decomposition Method in Subtraction
This method include borrowing an amount from a higher place value and giving it to a lower place value in the top number and re-writing the top number so that its original value does not change
As 7 cannot be subtracted from 3, borrow ten from the tens column and add it to 3 and the same time, reduce the number of tens from 6 to 5 as shown above, Now subtract 7 from 13 to get 6 in the units column. We cannot subtract 9 from 5 in the tens column so we must borrow 100 from the thousands column and add it to the tens column that is 10 to 5 to obtain 14 as shown above. It can also be seen that 4 has become 3. Now subtract 9 from 15 to obtain 6. The subtract 1 from 3 and 2 from 7 and put 5 and 2 to the thousands and hundreds unit. Hence, we obtain 5266.
Task 4 Measurement
At an early age, children are being taught of measuring the length, weight or capacity of an object, through the use of only language through adjectives like heavy, long, short and others. In this regard, one of the misconception of measuring length, weight and capacity is the students do not consider measure as part of mathematics and are constantly unable to make connections between measurement and other part of the subject. Such misconception is compounded by the notion that measures are often perceived as a discrete unit. As a teacher, it is important that we make meaningful connections in making sense of mathematics and measurement and making it easier learn.
In addition, another misconception of measuring length, weight and capacity is in line with using standard units. Because of the early knowledge that measurement is only measured in length by using descriptive words, they tend to forget that measurement requires exact and standard units. To enable learners to appreciate the needs for standard units, the students should work with pacing and other body measurements as part of their lessons. The students need to know that there is often a place for estimating before they measure it accurately and that accuracy of estimation enhances with practice. In addition, the teacher must teach children how to use measuring instruments in measuring the length, weight of objects and capacity of container. Furthermore, learners also need to be demonstrated how to read scales as well as dials. This is important to know how to set the scales to zero before they use it and they also need to use the information given on the scale like the actual numbers to count the division. All in all, children’s misconception of measurement can be corrected by helping them know the connection of mathematics and standard units.
Task 5 A. Is the following statement sometimes, always or never true?
If you roll two fair dice (numbered 1-6) and add up the total, it is just as likely to be 7 as it is 8.
It can be said that the statement is never true. Upon throwing two fair dice (1-6), there are 36 possible combinations that can be considered. In this combinations, there are 6 possibilities to get 7 (1,6; 2,5; 3,4; 4,3; 5,2 and 6,1). This means that there are 1/6 or .167 chance of getting 7. On one hand, there are only 5 ways to get 8 (2,6; 3,5; 4,4; 5,3 and 6,2), which means that there are 5/36 chance or .139 chances of getting 8. This further indicates that 16.7% > 139% probability of having 7 and 8, respectively. Hence, the statement is never true.
Task 5 A. Algebra
Identify strategies that children (across KS1 and KS2 age range) need to develop in order to recognise and understand about odd and even numbers.
Children can use various strategies in recognising odd and even numbers. One of these strategies is through the factoring method. The children can easily determine odd or even number if they are knowledgeable in factoring. Since odd numbers are numbers which cannot be divided exactly in two and the answer has always remainder which is 1 and an even number can be divided in two exactly, children will be able to determine whether a number is odd or even by knowing the factors of the number.
Task5 B.
This diagram could be used at the end of KS1.
When drawing the Carroll diagram the children should be very careful to make the boxes for information sufficiently big to accommodate the data. They can also forget to write the headings for each box e.g. red or not rectangle because they are so focused on sorting the shapes.
When interpreting the data, children might get confused and place all rectangles in one box and not notice that there is another place for rectangles that are a different colour. This is because it is all written in one vertical line.
Another problem that may arise is children forgetting to read the whole diagram and will omit writing information in the bottom left hand box.
Block Graph
These block graphs or bar graphs are normally used by pupils at the end of KS2
simon
john
hanna
silvia
semantha
January
108
45
88
58
37
February
24
39
65
92
75
March
75
35
12
85
25
April
55
41
74
24
17
One of the problems in drawing this block graph is that there is a lot of information on one single graph. Also it is very important to use distinct colours so that the data is easily identifiable.
As one is so focused on the comparison of the results of the various months for each child, it is difficult to compare the various children on the same month.
Task 5C.
Scatter Graph
This graph could be used in the second year of KS2:
Ten children took part in a sponsored walk. The scatter plot shows how much money they raised and how far they walked.
Top of Form
(a) How far did Karen walk? Miles?
(b) How much money did Karen raise?
(c) Who walked 20 miles and raised £40?
(d) How far did Bill walk?
(e) How much money did Bill raise?
(f) How much was Bill sponsored for each mile?
£ _______ per mile
(g) How much was Sunniva sponsored for each mile?
£______ per mile
One of the possible difficulties when drawing is to leave sufficient space in order to be able to plot all the information on the graph.
Another mistake would be not to take in consideration the fact that that the highest distance is 25 and to continue the graph until forty, in order to match the other axe, which will result in half the graph having all the data while the other half would be empty.
When a graph has similar numerical values on both axes, the pupil can mistakenly write the x coordinate on the y axis or vice versa.
Another common mistake would be to misinterpret the data as in one axes the value rises by two while in the other by one.
Task 6B
The key structures which are necessary for children to understand how to work out facts from known facts are:
“Using manipulative materials”
“Trial and error”
“Making a list”
“Drawing a diagram”
“Looking for a pattern”
“Acting out a problem”
“Guess and check”
:
“the integration of language arts as children write and discuss their experiences in mathematics.”
“children might draw pictures and then tell or write stories about the equation “
“Students can write a letter to tell a friend about something they have learned in mathematics class.”
Helping “children learn that mathematics is not simply memorizing rules and procedures but that mathematics makes sense, is logical, and enjoyable”.
“Creating and extending patterns of manipulative materials and recognizing relationships within patterns”.
Encouraging students “to explain their reasoning in their own words”.
Insuring that children “will not need to learn or memorize as many procedures; and will have the foundation to apply, recreate, and invent new ones when needed”.
Helping students to “view mathematics as an integrated whole rather than as an isolated set of topics”.
Task 7: Fractions
Accordingly, fractions are basic building block in the foundation of mathematics. In this regards, the teachers needs a good understanding of the topic so as to be comfortable in teaching students and performing operations with fractions and higher mathematics. For instance, fraction can be used in geometry to find the fractional parts of sets of polygonal regions. Herein, the context of adding and subtracting simple fractions is introduced utilising the pattern blocks. In this regard, the fraction are modelled as shaded part and students can create addition and subtraction problems with these. Furthermore, fraction can also be used in decimals as the students will learn that they can rename a fraction by multiplying its numerator and denominator to make decimal numbers. Hence, it can be said that fractions are useful in many areas of mathematics including algebra and other higher mathematics. In this regard, it is important that children must be able to be taught on playing with fractions from earlier stage of their learning for them to easily comprehend other used of fractions. As they consider higher mathematics courses.
Reference
Clements, DH (1999). Young Children’s Concept of Shape. Journal for Research in Mathematics Education, Vol. 30, No. 2, 192–212
Gelman,R . & Gallistel, C. R. (1978) The Child’s Understanding of Number (Cambridge, MA, Harvard University Press).
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