Literature+Based+Responses+to+Identified+Issues+and+Challenges

**Formation: Issues and Challenges**
Miscalculating or not knowing number facts can have an affect on working out prime and composite numbers and prime factors.

**Response**
Use a number board with students and get them to count individually as a transition. Practice with learners as much as possible in every learning experience enabling them to understand the importance of sequential numbers and patterns. write numbers with children on a board or in a workbook and make a game of it - play based learning is always a way for students to take in an activity and use this knowledge learnt to apply it to mathematical situations. "Take as many opportunities as you can (as an educator) throughout the day - does not have to be within a maths lesson to count with the children. Model numbers to learners and use counting strategies to find and verbalise relationships between small numbers." (Tertni, J. 2010) Teach and learn mathematics as a language and children will be able to understand how to justify an answer. Using the teaching method of modelling will assist the students comprehension and rub off on their understanding to a learning experience.

**Year 1: Issues and Challenges**
Counting to and from 100 - missing numbers, not knowing the tens column numbers (what comes next), counting in 2's, 5's, 10's - a pattern has to be understood before students will understand how to do this, writing numbers correctly (no backward numbers, understanding place value with digits which column do the numbers belong to? Connect number names, numerals and quantities, including zero, initially up to 10 and then beyond (ACMNA002)

**Response**
"How do we teach counting? Start with patterning. Just say the numbers from one to five, then one to ten then ten to twenty. Repeatedly." (Ramone, C) Children respond to familiar practices and routine, this comes down to perseverance in your own teaching. "every job presents challenges however being self confident about your ability to seek answers and solutions may increase your willingness to accept challenges." (Marzano, R. Pickering D et al. 1997) Students will also learn with concrete materials to group and count in twos (such as counters and a number board - odd and even numbers; colour coding digits specified in the lesson plan) and these chlalenges will be overcome by the learners in your class.

**Year 2: Issues and Challenges**
Recognising and understanding number patterns, justifying why there is a pattern with numbers may be a challenge for students if the concept is not fully understood, addition and subtraction common challenges - counting on and countin gback needs to be known effectivley, grouping - are the groups split equally? getting to know the innovative mathematics language.

**Response**
There are many challenges to teaching addition and subtraction as there are many ways to teach it. All students have different ways of learning and interpreting what is being taught. In the early years, when teaching adding and sutracting there are many games that can be played with learners to keep their minds engaged. This may include activities such as dominoes or dice games (by using interactive resources the students are encouraged to include themselves in a learning experience). It is important for you as a teacher to use the correct materials when teaching for activities to be successful in order to achieve a goal. "being aware of necessary resources involves a cycle of assessing what resources are needed, determining their availability, accessing them, using them appropriatley, and continuing to reassess them to identify additional resources as you work" (Marzano & Pickering. 2009) The materials that are used in a learning experience can assist in solving problems with confusion and comprehension of a maths task for a student.

**Year 3: Issues and Challenges**
Writing and spelling numbers correclty - children should be able to use their knowledge of phinics although will need to learn to write numbers, learning maths as a language.

**Response**
Mathematics is not only a concept with numbers it is also a language. Children need to be taught how to justify and anwer to a problem then explain it mathematically. "Our math education system pays some attention to the idea that math is a language. For example, many math teachers have their students do journaling on the math learning experiences and their math use experiences. Some math teachers make use of cooperative learning--an environment that encourages students to communicate mathematical ideas. Some math assessment instruments require that students explain what it is they are doing as they solve the math problems in the assessment" (Moursund,D. 2012). There is a need for students to be able to justify their answers in a mathematical way. For students to learn this effectivley it is important for you to teach every concept correctly. Learning to spell numbers and stretch a number when it is longer than one word is also important. Students are to learn phonics and english concepts in maths. This challenge can be overcome by displaying numbers and the words for numbers around the classroom allowing students to constantly view and become familiar with numbers as words.

**Year 4: Issues and Challenges**
Getting place values confused - ones, tens, hundreds, thousands and millions. Confusion with how to read large numbers.

**Response**
ʺ While it is difficult to conceive of or appreciate the size of large numbers, their frequent usin in modern society meeans that children must deveop and awareness of their meaning ʺ (Booker, Bond, Sparrow & Swan, 2004, p. 125). Booker et al. also discusses how visualising large numbers and quantities is a difficult task, the best way to understand theses large numbers is by understanding their order by interpretting the way they are written, as well as how they are spoken. As learnt in earlier grades, there is a set pattern of ones, tens and hundreds. To correctly name larger numbers, continuation of this pattern is essential.

The ACARA content descriptor for year four states that students should be able to ʺ Apply place value to partition, rearrange and regroup numbers to at least tens of thousands to assist calculations and solve problems (ACMNA073) ʺ. Booker et al. state that ʺ Children need to investigate numbers to millions or beyond to fully comprehend the way in which the place value pattern is extended to large numbers. Simply stopping with the hundred thousands and then 1 million leaves them uncertain as to how to deal with lage numbers... ʺ (2004, p. 127).

The pattern in numbers of: the first three digits on the right show hundreds, tens and ones of ones, the next three digits show hundreds, tens and ones of thousands and the next three digits show hundreds, tens and ones of millions is continued onto billions and beyond. Number expanders are a perfect tool to assist learners in reading large numbers as they "show the total number of thousands hundreds, tens or ones rather than just the digit in each place". (Booker et al. 2004, p. 126).

Siemon et al. (2011, p. 309) state that "Regular opportunities need to be provided to compare, order, count forwards and backwards in place value parts, and rename numbers..." and that as well as number expanders, number lines are ideal for illustrating counting forwards and backwards. Siemon et al. also accentuate the use of the word 'makes' to assist in understanding place value. For example, ten $10 notes makes $100.

**Year 5: Issues and Challenges**
Poor rounding and estimating skills might lead to major errors in problem solving.

**Response**
Using the terminology rounding 'up', 'down' or 'off' can cause confusion to students who are begining to learn about rounding. Booker et al. describe rounding 'up' or 'down' simply as a matter of choice for students as they simply follow the rules on rounding and have no understanding why it is done this way. It is deemed mathematically valid and less confusing for students to talk about 'rounding to the nearest' instead of 'up' or 'down'. "Rounding numbers to the nearest specified place-value part can be difficult for children if they rely on remembering a rule or procedure, particularly where this is couched in terms of 'up' and 'down'". (Siemons et al. 2011, p. 315).

Utilising a 0-99 board is one simple way to highlight the main idea of rounding to the nearest ten. Generally, a number ending in 5 is supposed to be rounded 'up' and the 0-99 board clearly shows why this is so. "There are ten numbers in each row. Five numbers round to the tens they have and four of the others clearly round to the next ten". (Booker et al. 2004, p. 121). This concept is easily demonstrated on the 0-99 board which gives the students an understanding of the generalised rule of why 5 rounds 'up' and not 'down'.



Another issue involving rounding is the scenario in question - this will assist in determining which place value the number needs to be rounded to. For example, rounding the total number of people living in a town can probably be rounded to the nearest thousand (62 783 people is approximately 63 000 people), but on a grocery bill $59.88 should be rounded to th $59.90 instead of $60.00.

Booker et al. (2004, p. 181) point out that children being able to instantly recall number facts is not sufficient when answering mathematics problems. Estimation is one way for students to check the reasonableness of an answer. For example, if they are entering a sum into a calculator, they need to be able to double-check the answer provided to them. If the student is required to calculate 41 + 49, they could use their knowledge of rounding to make this sum 40 + 50. Although estimating does not usually give a completely accurate answer, it gives the student a number which is close to the answer so that when they do have an answer to the original sum, they are able to identify if their final answer is reasonable or not.

For a student to be able to estimate the answer to a mathematical problem, Siemons et al. (2011, p. 388) believe that the student "requires a well-developed sense of number, a knowledge of number facts and relationships, and an understanding of the meanings of the operations".

**Year 6: Issues and Challenges**
Misunderstanding of problem solving; using incorrect order of operations in problem solving.

**Response**
Booker et al. (2004, p. 51) discusses some major issues that students have difficulties with when attempting problem solving. These issues include that students simply: located and manipulated numbers; tried different operations; looked for keywords; read problems quickly and cursorily; gave no consideration to the problem context; did not assess the reasonableness of answers; showed little perseverance if answers could not be obtained from the first approaches tried; did not access problem solving strategies even though they could generally be named and described when discussed as entities in their own right. Schoenfeld's observations in 1992, (as cited in Siemons et al. 2011, p. 78) support these issues by stating that "students learn 'that answers and methods to problems will be provided to them ... or may curtail their efforts after only a few minutes'". Encouraging students to think mathematically is a vital element for teachers to use when structuring problem solving questions and problems. It is necessary that questioning and justification is included in in task planning (Siemons e al. 2011). As identified above, trying different operations is a common strategy used by students in problem solving. Booker et al (2011) elaborates on this by saying that students try different operations and order of operations if their original answer did not seem reasonable. This strategy identifies that the students do not understand what the problem is asking them to solve and that the students are simply locating and manipulating the numbers at a surface level. Students need to learn strategies like re-reading worded problems to locate the necessary information to answer what is actually being asked. Checking answers for reasonablness is another strategy which can be used to assist students in problem solving. Checking for reasonableness can often allow the students to identify if they have used and an incorrect operation (multiplying instead of adding). As the concept of estimating has been introduced at previous year levels, by year 6 students should have a sound understaning of estimating answers by using number facts and strategies. Booker et al. (2011) also discuss the strategies of trying possible solutions, working backwards, using materials and drawing diagrams when exploring problem solving. Regarding order of operations, students need to understand that "When working with mixed computations, then there are three rules that allow answers to be determined unambiguously:
 * 1) First complete any calculations within brackets.
 * 2) Then do any multiplication or division before addition and subtraction.
 * 3) The work from left to right". (Booker et al. 2011. p. 362).

When calculators are used in problem solving, students purely enter the numbers and symbols in the same order they are written which completely disregards the three rules mentioned above and will inevitably result in the calculator providing an incorrect answer for the problem. Mathematic problems written in words and sentences also adds to the challenges that students might face when solving problems. The terminology used may be unfamiliar; unnecessary information is included with the necessary information making the problem seem more difficult than it really is; and sometimes not enough information is provided.

**Year 7: Issues and Challenges**
Confusion differentiating which law applies to different sorts of mathematical sums.

**Response**
When reading the definitions of the associative, commutative and distributive laws, they sound easy to understand, but when it comes to applying these laws to different mathematical situations there is quite a lot of background knowledge and understanding required by the students to correctly choose and implement the laws.

The commutative law is probably the easiest law to understand and explain, and students in year 7 would be familiar with this concept although as Booker et al. (2004) says, this is not the terminology that students would be used to hearing or using. The commutative law is that when multiplying or adding numbers. Student would recognise this as 'turn-arounds' where 10 x 3 is the same as 3 x 10, and that 25 + 35 will give the same answer as 35 + 25. It is sometimes difficult for students to understand that this law only applies to multiplication and addition, but not to division. In other words, if you performed 12 / 3 = 4 and attempted a turn-around of 3 / 12, the answer would be 1/4 which is not the same answer as the original operation. Therefore the commutative law does not apply to division (Pierce, 2012),

The associative law expands on the commutative law. With addition and multiplication, the associative law says that it does not matter how you group the numbers:

(a + b) + c = a + (b + c) //(addition)// and (a x b) x c = a x (b x c) //(multiplication)// As with the commutative law, the associative law only works with multiplication and addition - it does not work with subtraction. For example, (10 - 3) - 2 = 7 - 2 = 5 but 10 - (3 - 2) = 10 - 1 = 9 which is a different answer form the original operation, which proves that the associative law does not apply to subtraction (MathsIsFun.com, 2012). The associative law can be used to:

Add or multiply in a different order - 21 + 3 + 7 = 21 + (3 + 7) = 21 + 10 = 31 or to re-arrange the numbers a little - 2 x 9 x 7 = 9 x (2 x 7) = 9 x 14 = 126

The distributive law is written as: a x (b + c) = a x b + a x c

This means that you get the same number when you multiply a number by a group of numbers added together as when you do each multiplication separately. Therefore, 3 x (2 + 4) will give the same answer if the sum is written as 3 x 2 + 3 x 4.

Pierce (2012) gives many examples as to when the distributive law can be applied: Sometimes it is easier to break up a difficult multiplication: Example: What is 6 × 204 ? 6 × 204 = 6×200 + 6×4 = 1,200 + 24 = 1,224 Or to combine: Example: What is 16 × 6 + 16 × 4? 16 × 6 + 16 × 4 = 16 × (6+4) = 16 × 10 = 160 You can use it in subtraction too: Example: 26×3 - 24×3 26×3 - 24×3 = (26 - 24) × 3 = 2 × 3 = 6 You could use it for a long list of additions, too: Example: 6×7 + 2×7 + 3×7 + 5×7 + 4×7 6×7 + 2×7 + 3×7 + 5×7 + 4×7 = (6+2+3+5+4) × 7 = 20 × 7 = 140

Confusion surrounding the distributive law may occur because this law does not apply do division problems.

**Year 8: Issues and Challenges**
Not understanding index notation and index laws.

**Response**
Victoria University (2012) explains index law as "Some numbers can be written in mathematical shorthand if the number is the product of 'repeating numbers'. eg 100 is the product of 10 multiplying itself two times: 100 = 10 x 10 or 64 is the product of multiplying itself six times. 64 = 2 x 2 x 2 x 2 x 2 x 2". To display these numbers in mathematical shorthand, they are expressed as 100 = 10 2 and 64 = 2 6. In this shorthand method, in 10 2 the 10 is called the base number and the 2 is called the index number. Likewise with 2 6, 2 is the base number and 6 is the index number. Index form can also be referred to as index notation and power notation.

Exponents are another way of describing index numbers. Pierce (2012) simply defines using exponents as "The exponent of a number says how many times to use that number in a multiplication". To summarise: "8 2 could be called "8 to the power of 2" or "8 to the second power", or simply "8 squared".'" (Pierce, 2012).

Simplified,a power of a number (or the index) indicates how many times that number needs to be mulitiplied by itself.

Work covered by index notation will probably incorporate prime numbers, factor trees and prime factors and ths work lead into scientific notation which is learnt in later years. Booker et al. (2004, p. 132) states that "Difficulties with numbers written using standard form or scientific notation usually occur because of a lack of understanding of the concept of factor and the use of exponents in writing the product of several of the same factor".