Patterns and Algebra

Issues/ Challenges

Solutions/Strategies

Justification/Origin

Prep

1. Students may be unable to recognise when objects have been organised into a pattern.

Teachers need to explicitly teach students what the term 'pattern' means and model to them how to recognise and create patterns.

Students also need their attention directed to the fact that patterning is everywhere around them in the world (E.g. seasons, traffic lights, petals on flowers, school day routines, rhymes, stripes on a zebra crossing etc.). Learners can discuss as a class about the different types of patterns they have seen in their day to day life and could take a trip around the schoolyard to investigate all the patterns they can find. This helps to put the concept of patterning into a meaningful context for students.

Picture books involving numbers and patterning can also be a very useful tool for helping students to recognise patterns in the world around them. For example, 'The Patchwork Quilt' by Valerie Flournoy introduces students to the way patterning is used to create different effects on quilts. The students could talk about the different types of patterns that have been used (Eg: ABAB) and what features make up the different patterns (Eg: Shapes and colours).

It is crucial that students develop structural and patterning thinking in maths during the early years so that they are able to recognise mathematical structure, and more importantly, understand how the number system is organised by grouping in tens (Mulligan, Prescott, Papic & Mitchelmore, n.d.).

Origin: Children who do not actively look for patterns around them as a way of understanding and learning mathematics are often those who find math to be very difficult. This is why it is crucial that during the first years of schooling children are explicitly taught how to look for patterns, create patterns and extend patterns (New Jersey Mathematics Curriculum Framework, 2006, p. 338).

Young children need to have their attention focused on the way that patterns are a part of their daily life. Patterns are in songs, rhymes, artwork nature and day-to-day routines (IECER, 2012).

2. When copying or continuing a pattern, learners may only replicate a certain part of the original pattern (Eg: Original Pattern = ABC. Student may continue the pattern like so, ABCABABAB, leaving out the 'C' part of the original pattern.).

Learners must be specifically taught that patterns (repeating ones) are repeated over and over again based on the exact original pattern given. Teachers can model this to students by constructing a repeating pattern on the board and highlighting the way that all of the objects/shapes/letters have been repeated exactly the same as the original pattern each time.

Another strategy to help learners understand that the whole of the original pattern must be repeated when creating patterns, is to have the learners act as 'error detectives'. The teacher writes/draws a number of patterns on the board and the students' job is to search through the patterns to see if a repetition error has been made.

To practise their understanding of patterning, students can work to create their own repeating pattern using a variety of hands-on materials such as unifix cubes, shape cut-outs, toothpicks and coloured counters. Learners may also like to work in small groups where one child creates an original pattern (green counter, red counter, blue counter) and the other has to continue the pattern using the same objects.

One of the most common errors that learners make when copying or creating patterns is skipping parts of the design that were meant to come next (Lee, 2011).

Origin: This is often because children only repeat the element of the original pattern that they saw last. Students may be able to repeat the pattern successfully a couple of times but then lose track of where they are up to and add in objects that do not belong in the pattern (Lee, 2011).

Students might need intervention to help them recognise what part of a pattern is repeating. Often children can identify part of the repeat but not the whole repeat. Teachers need to encourage learners to stop after each time they make a repeat to check that the pattern is correct before moving on (Siemon et.al., 2011).

3. When learners are asked to sort objects into groups according to their different attributes, students may have difficulty identifying the features that help the items to be sorted. Furthermore, students may be able to sort the group of objects correctly, but may be unable to communicate why they sorted the items the way they did.

Teachers need to prepare students for sorting/classifying tasks by explicitly showing them some of the features that help different objects to be grouped together - such as colour, shape, size and texture. The teacher could have a shared learning time where they have a whole number of different objects spread out on the floor and the students have to give ideas for how the items might be grouped, as well as communicate reasons why they could be grouped that way (teacher uses questioning to help prompt the learners thinking). Through allowing students to share their ideas but requiring them to communicate the reasons behind their thinking, shows students that they cannot just 'do' an activity in Maths, they need to think carefully about the choices they have made and be able to explain why.

ICT: To help scaffold the children's ability to recognise the different attributes by which objects can be sorted, the teacher could initially provide them with a sorting activity where the features are already given. This might be in the form of an interactive online sorting game where the students need to drag different coloured socks into the baskets labelled 'Red Socks', 'Blue Socks' and 'Green Socks'. When students have achieved success in the interactive activities that have the features provided, they can move on to another web-based sorting game that does not give the learners the features by which the items need to be sorted. This website contains a number of effective web-based sorting activities that would be suitable for this lesson -http://karenogen.blogspot.com.au/2011/01/sorting-and-classifying-objects.html

Sorting can be one of the hardest concepts for children to grasp in the early years, particularly when children have to attempt to sort objects that have more than one attribute (Eg: Colour and shape). Teachers often find that their learners are able to successfully sort by colour, but when it comes to other features like size and texture they have difficulty unless they are told what feature to sort by. This is why it is important for teachers to encourage learners to sort objects into groups by attributes other than colour (Lee, 2011).

Learning mathematics in the first years of schooling requires more than giving children concrete materials, such as blocks or coloured markers, to explore with. This is because math is ultimately about thinking, not just doing things with objects. Teachers need to help students cement what they have learnt while 'doing' through using appropriate vocabulary and questioning to guide children to reflect on what they have learnt (IECER, 2012).

Year 1

4. When skip counting, learners may have difficulty beginning from a number other than the starting number of the sequence (Eg: Learners may find it easy to start from the number 2 when skip counting in 2s, but may have difficulty if they are asked to start counting in 2s from a number such as 14).

Students need to have opportunities to not only skip count from the start of a number sequence (Eg: Counting from the number 2 when counting in 2s), but to skip count from numbers with much higher values and in different directions (forwards and backwards). When teachers only ask their students to skip count from the number 2 to the number 20 when skip counting in 2s for example, students may find it very difficult to count in 2s beyond the number 20. Through only practising skip counting by ROTE and not giving learners opportunities to skip count forwards and backwards from multiple starting points, they will not develop the fluency needed for more difficult multiplication and division sums later in schooling.

Learners could complete a number of different activities that help to practice skip counting other than the use of ROTE learning. These might include: giving each student a number chart where they have to colour in the numbers they would use to skip count in 3s from the number 34 (34,37,40,43...) or providing the learners with a roll of paper (like those used for cash register receipts) and they have to choose their favourite number (Eg: 23) and skip count from that number in 4's, writing their number sequence vertically down the paper roll.

ICT: Learners can use a digital based hundreds chart for this activity which can be found at this website - http://www.abcya.com/interactive_100_number_chart.htm

Young children need support to develop fluency with counting forwards and backwards by different numbers (Eg: counting in 2s, 4s and 7s) and from multiple starting points (Perspectives - An Administrator's View, 2012).

Origin: If students cannot fluently count in these ways they will find it very difficult to solve multiplication and division problems in later grades (Perspectives - An Administrator's View, 2012).

There is great value in drawing children's attention to the patterns involved with counting by different numbers. It supports their sense of number in general and helps to prepare them for understanding the concept of divisibility (Siemon et.al., 2011).

5. Often learners have difficulty recognising the different number patterns that occur when counting to 100, such as the occurrence of the digit '0' with each new group of tens. This can cause students to count past the number 29 saying 'twenty ten' instead of 'thirty'.

The digit zero ('0') is often a number that children have difficulty with when counting because it is used to represent each new group of tens. When learners are beginning to count beyond the numbers 20, they can sometimes apply the same counting pattern they used when counting from 0-20. This results in students naming numbers like 'twenty-ten' when counting on from the number 29, instead of 'thirty'. Learners who have this misconception need to be explicitly taught about the fact that the '0' in numbers indicates that a new set of 10 numbers is about to be counted. Essentially, students need to know that the pattern used to count the teen numbers is special. Teen numbers are only used in counting between the first twenty numbers of every sequence of 100 numbers.

A strategy for showing learners this concept in a concrete form is the use of 'tens frames'. The teacher could show the student the number twenty on two tens frames. They would then go on to explain how 'twenty-ten' cannot be a real number, because it would mean adding ten more counters to the next tens frame and that would equal thirty (teacher would have the student count the lots of tens so they could physically see that the groups of tens equal thirty). This would help confirm to the learners that the number they were naming as 'twenty-ten' is in fact thirty.

Children will often recite numbers they have invented (unknowingly) when counting, such as 'twenty-ten'. This is not a number the child has heard being spoken by teachers or parents, but is a number they have constructed based upon the patterns they know occur in the counting sequence. In the child's mind, ten follows nine and so it seems completely sensible that twenty-ten must follow twenty-nine (Baroody & Wilkins, 2004, p. 50).

Year 2

6. Students might be able to successfully skip count in 2s, 5s or 10s, but may not be able to describe the pattern created when skip counting (Eg: When counting in 2s, 2 is added to every new number).

Teachers need to explicitly teach students that skip counting is not just counting up or down in different numbers, but that the numbers being counted up or down actually form an addition or subtraction pattern (Eg: Counting up in 2s is really adding 2 to each new number). Helping learners to be able to see and describe the pattern created when skip counting has a lot to do with the language and means by which the teacher introduces the concept. If the teacher simply tells the students that they are going to learn how to count up in 3s and the learners then practise doing this by ROTE, they will not have be able to establish that they are adding 3 to each new number they count.

To ensure that this misconception does not occur, teachers need to use concrete materials to assist the students to see the addition patterns that are formed. For example, the teacher could give a learner two marbles and then ask them to add another two marbles to the pile (4) and then add another 2 marbles (6) etc. The teacher would then ask the learner what they did each time to the pile of marbles (added 2 marbles) and then connect this to fact that when they are skip counting in 2s they are adding 2 numbers to the one they just counted, like the marble example. The use of hundreds charts and number lines when teaching children to skip count are also effective tools for helping them to clearly see the patterns made.

ICT: Students can also use their calculators to investigate the different patterns made by skip counting as the numbers increase by a constant amount when the equals sign '=' is pressed. The teacher might type in 4 + 3 and the learners have to guess what the next numbers in the pattern will be before they press the '=' button.

Students need to be able to investigate and describe number patterns formed by skip counting and patterns with objects (ACARA, 2012).

Using a hundreds chart when teaching young children to skip count is very effective in demonstrating number patterns. By shading the numbers as they skip count, children can clearly see how a certain number of squares are either added or subtracted to each new number. As the hundreds chart is a visual diagram, learners can also use it to predict what number might be next in a pattern - Eg: "If you count by tens beginning at 36, what number would you colour next?" (PSSM, 2000).

7. When solving word problems, students may have difficulty trying to determine what operation they need to perform (addition or subtraction) from the information provided. Eg: Sally went to the shop and bought six bananas. Her friend Addison came over and Sally gave her two bananas. How many bananas does Sally have left? Operation = subtraction. Learners may not recognise the language cues that help them to determine what operation they need to perform.

Learners need to be explicitly taught the different vocabulary used to represent addition or subtraction in word problems (Eg: 'How many altogether' = addition, 'How many left' = subtraction). The teacher could construct a word wall that shows the different types of words that the students might meet in both addition and subtraction word problems. This would act as a reference point for learners when completing these types of problems to assist them with determining what operation is being used.

Another strategy that students can use to help solve word problems is by drawing a picture to represent the sentence. For example: 'Noah bought 12 apples from the shop and ate 3 for breakfast on Monday and 2 for breakfast on Tuesday. How many apples did Noah have left for the rest of the week?'. The learner could draw a picture of 12 apples and then cross out the three that Noah ate on Monday, and the two that Noah ate on Tuesday. Then they would simply count the apples that were not crossed out to find the final number of apples Noah had left.

To practise their use of both writing and solving word problems, students could work in pairs to construct their own word problems for a peer to solve. The learners would then swap the word problems they have written, answering their partner's problem and highlighting the words that helped them to know what operation to use.

Origin: Word problems are often one of the hardest math tasks for children to complete because they have difficulty trying to decide which operation the problem requires. This often results in students applying 'coping' strategies to try and make sense of the problem, such as adding whatever numbers are present in the sentences regardless of what the words are telling them (Shwarz, 2005).

Origin: Being able to choose what operation is used in a word problem is a crucial part of learning how to turn English sentences into mathematical expressions. Students might have the mathematical skills to solve the word problem, but if they cannot understand the literal meaning of the sentence they will be unable to progress any further. For learners to be able to successfully solve word problems they need to use both reading and mathematical skills. Further to this, children need to have a clear understanding of the language cues that suggest what operation is being used (TeacherVision, 2012).

Students need to be able to solve problems by using number sentences for addition or subtraction (ACARA, 2012).

Year 3

Students have difficulty identifying the type of pattern (repeating, increasing/decreasing patterns) and then applying the rules to patterns

As a class the LM demonstrates to students an increasing pattern, through the use concrete materials or displayed on the interactive whiteboard.

Then in small groups students rotate around the room creating their own patterns using different concrete materials such as, paddle pop sticks, counters, tiles, blocks, base ten blocks etc. Students and LM can take photos and record down groups different patterns. Diagnostic testing can occur to identify any prior knowledge.

LM can again use coloured counters, blocks and picture diagrams as patterns. LM starts a shape pattern:

Students then continue the pattern.

Instead of represented in shapes, numbers are substituted.

ICT: Students interact with a number line produced on the interactive whiteboard can also help students with visually identifying how many numbers were missed in the pattern and then identifying the rule.

For example: LM highlights particular numbers on whiteboard and students then count how many numbers were missed. Then continue the pattern using the rule.

Origin: It is common for students to confuse a repeating pattern with an increasing or decreasing pattern (Mathematics Curriculum Guide, 2010, p.34).

Students need sufficient time to explore increasing patterns through various mediums, such as link-its, tiles, flat toothpicks, counters, pattern blocks, base ten blocks, bread tags, stickers, buttons, etc., to realize they increase or decrease in a predictable way. Later, students will connect patterns to numbers, and work with patterns found in the hundreds chart or record patterns in a T-chart (Mathematics Curriculum Guide, 2010, p.34).

Clear links are made to the ACARA (2012, p. 26) Number and Place Value sub-strand: Apply place value to partition, rearrange and regroup numbers to at least 10 000 to assist calculations and solve problems. In order for students to construct knowledge place value should be considered as foundation or prior knowledge, which patterning is then built upon.

Through the use of the interactive whiteboard students are actively engaged within the learning, which is important when learning mathematical concepts as they are traditionally repetitive (SMART Technologies Inc., 2006 p. 4).

Diagnostic assessment for the learning of patterns is helpful when understanding student abilities with solving pattern problems (Feifei, 2005, p.3).

Year 3

Continuing patterns in both forward and backward directions is a challenge, due to the fact that students bring their prior knowledge of reading from left to right into the context of mathematics.

As a class students create the beginning of an increasing pattern or a decreasing pattern. Then in pairs or groups they continue the pattern backwards. After this the LM will provide an opportunity for individual mastery.

For example:

OR

Number lines and hundred boards are also effective tools that allow the directions of numbers to be visually displayed.

Students are required to be confident with place value to understand that patterns can go in both forward and backwards directions. Therefore, if students are having difficulty, the use of concrete materials, such as base ten blocks, provide students with a visually and kinaesthetic experience to understand if there are, now, more or less of the number pattern. These hands on materials should also be used in conjunction with explicit teaching of place value.

Origin: Students are more likely to make mistakes when completing subtraction (Koshy, Ernest, Casey, 2000, p. 33 -34) and therefore continuing patterns in a backwards direction, which involves subtraction, can be difficult.

Fuson and Briars (1990) reported astounding success in the use of base-ten blocks in teaching addition and subtraction algorithms, cited in Thompson, (1994, p.3).

By using real object representations, young children are capable of masterful algebraic thinking.(Taylor-Cox, 2003, p.18)

Patterns involving multiplication and division functions are learnt through rote which results in a challenge for students to develop in-depth understandings.

Rote learning is a good tool for learning multiplication and division facts, but it is important that it does not become the sole means by which learners construct understanding about multiplication and division.

Possible problem solving situation: You are working at Celebrate in Style, a decorating business. A customer comes in and notifies you she is having a party for 32 guests. How many tables will be required for a sit down meal? As a class, students can problem solve this real life contextual problem. Students create a pattern to show the layout of tables and chairs. Students are then told that the room is triangular in shape and that some kind of pattern is needed so that the tables are in formation.

LM again uses a hundreds chart on the interactive whiteboard and allows the students to highlight numbers a particular colour and visually see if some numbers can be divisible by more than one number. For example, the number 12 can be divided into 6, 2, 4 etc.

ICT: An engaging way for students to become familiar with their multiplication facts is through participating in an online timed game called 'Matho'.

URL: http://aplusmath.com/Games/index.html

Origin: Rote learning is still a prevailing technique of teaching in many schools (Norman, 2008). Norman (2008) continues stating that some knowledge is appropriate to learn by rote however most are not as the students need deep understanding so that they can then apply the learning to a variety of new contexts.

A common result found when rote learning is used is that students often dont know how to fix their errors (Norman, 2008). Therefore ensuring students comprehend the multiplication process is paramount when dealing with patterns, as this process will be manipulated.

A clear link is made to ACARA (2012, p. 29) other sub-strands, Number and place value: Investigate number sequences involving multiples of 3, 4, 6, 7, 8, and 9 (ACMNA074). This should be introduced to ensure continued knowledge can be developed involving multiplication number patterns.

Including a real-life related context in which students can use this information will ensure a purpose to their knowledge (Marzano & Pickering, 1997, p.30).

Students must come to see and believe that mathematics makes sense, that it is understandable and useful to them. They can become more confident in their own use of mathematics (Daniels, Hyde & Zemelman, 2005, p.32).

Year 4 and 5

Use of subject-specific language such as the terms 'array' and 'equivalent' may be new to students and need to be defined as soon as learners meet those words in the learning sequence.

Constructing a word wall of the specific language required for different mathematics units is a great strategy to ensure students have a reference point if they are unsure about any of the vocabulary being used by the teacher to describe different concepts.

Providing students with group tasks will encourage discussion and provide an opportunity for students to exercise the mathematical vocabulary they have learnt to share their ideas. This will help them to become more confident with the language of mathematics and pick up on new terminology from listening to their peers.

Origin: Hutnick states that mathematics has its own language and becomes easier when students and teachers can speak that language. (2004, p.13)

Talk-based group activities can help the development of individuals mathematical reasoning, understanding and problem-solving (Mercer & Sams, 2006, p.1) Therefore, LMs are required to provide a universal language that students can communicate with and grasp collective understandings.

Education world (2011, p.1) explains the benefits of Word Walls stating, "Seeing words on the wall helps them become excited about words and understand that words are important and can be used over and over again.

Year 4 and 5

In equivalent sentences and number patterns the equals sign '=' does not represent the answer, but rather shows that the sentence is balanced.

Scales are usually used for measurement, however can be also used as a visual representation to students when finding unknowns in equivalent number sentences and patterns.

Jenifer Taylor- Cox (2003, p.5) provides an example detailing the use of scales in an algebraic setting. Students are given this problem:

There are six children in the Can family (each represented by a film canister) arrive at the park. All six children want to ride the super seesaw at the same time. To help the Can family decide who should ride together, place the Cans on the scale to show balance. Draw a picture of what you find out (Taylor- Cox, 2003, p. 5)

Six film canisters with varying amounts of sand inside were given to students. Two canisters are filled completely (A=Alex and A=Angela); two are filled halfway (B=Burt and B=Belinda); and two have only a few grains of sand (C=Cory and C=Caitlin).

The students would then be given a problem to solve such as: If Alex was to get off the seesaw, would the seesaw remain balanced? Through working with the scales the learners are be able to see that the seesaw could only remain balanced if Angela got off as well. Therefore the equation for the problem would be written, BC = CB.

This example links directly to algebra as students will understand that for sentences to be classed as equal/equivalent both sides of the equal sign or scales needs to be the same.

According to Jenifer Taylor-Cox (2003, p. 5), scales are a key tool that help students understand that for number sentences to be equal, the values must be the same on each side of the equals sign '='.

Origin: Early intervention and understanding of what equivalent means will limit further challenges with more complex algebraic algorithms (Welder, 2008).

Year 4 and 5

When students are creating patterns using decimal numbers, they may encounter difficulty if they do not have a clear understanding of the decimal place value system (tenths, hundredths and thousandths).

Money is a fantastic context for LMs to teach decimal place value and number patterns, because it involves dollars and cents (dollars = whole numbers and cents = parts of the whole number/decimal numbers). For example, students could be given the questions: How many 20c pieces make up $1? How many will make up $6? Using real coins, the students can model the number pattern and then create their own challenge questions for a peer to work out using the coins.

After introducing decimal place patterns within the context of money, more complex patterns can be explored using an interactive whiteboard software program.

The math software program 'Hi-Flyer Decimals', uses MAB blocks both for whole number systems and for decimals to represent place value. This is interactive for students and caters to a variety of learning styles with audio, visual and Kinaesthetic learners. Furthermore, LM can use this as a diagnostic tool URL: http://www.classroomprofessor.com/teaching-math/software-decimals/

This challenge originates from whole number place value system. Students assume that thousandths is the larger number, because thousand is larger than hundreds and tens. Therefore, a confident foundation of knowledge about directionality is required for students to construct this new knowledge (Moody & Piffath, 2005).

Allowing students to physically interact with the board can assist with meeting the needs of tactile learners (Beeland, 2002, Cited in SMART Technologies Inc., 2006 p. 9).

A good understanding of place value of whole numbers and its extension to decimal numbers is vital because it is the basis of both our mental and written calculations Casey (2000).

Year 6 and 7

Transitioning from concrete patterning to viewing patterns as functions (Warren, 2005).

Example: Student attempts to create the pattern of 2r + 3b (r= # of red counters and b=# of blue counters) using concrete materials but quickly come to the conclusion that it is too large to model. This step requires students to know about functions and how they can be used to represent patterns.

Students will solve 4 or 5 equations that vary from one step to multi-step. Students will solve for x mentally, and then describe a method for showing their work algebraically.

Explicit teacher guided activities where students are able to attempt to record concrete patterns in algebraic function form.

Group students into pairs, one partner is to create a function that is representative of a basic growing pattern, the other student is to create a pattern using concrete materials. These two students swap places and complete their partners problem, one needs to make the pattern from a function using concrete materials and the other has to form a function from the pattern their partner made.

Model on the board to students how a function can go to a pattern and then back to a function.

Create a Math a Pair activity where each student is given a card with either a function or a visual pattern on it. Students are to walk around talking to their classmates to find the card that matches theirs.

Identify patterns that can be found in students lives and use a function that has been derived from these to predict future outcomes.

Origin: Often attributed to reasons such as students having never developed the conceptual understanding of the target mathematics concept/skill (USF, n.d.)

Conceptual understandings of a target mathematics concept/skill is demonstrated by the ability to recognize functional relationships between known and unknown, independent and dependent variables, and to discern between and interpret different representations of the algebraic concepts. It is exemplified by competency in reading, writing, and manipulating both number symbols and algebraic symbols used in formulas, expressions, equations, and inequalities (Panasuk & Beyranevand, 2010).

Scaffolded questions allow students to review and build upon their prior knowledge (International Centre for Leadership in Education, 2007).

Students need to first be taught that letters can be used to signify numbers. These letters are then to be used in number patterns and finally extended to number patterns where a function or rule needs to be derived (Booker, Bond, Sparrow, & Swan, 2010).

A thorough understanding of the general concepts associated with patterning and algebra provide students with a richer background to algebraic thinking (Booker et al. 2010).

ACARA Links: The Number and Algebra strand has strong links to this challenge in that students will be using whole numbers, and all four operations when creating and solving their functions.

Year 6 and 7

Identifying and articulating the missing steps in an incomplete pattern (Choi, Huh, LaRue, 2010).

Example: Students may be able to continue a pattern when the first three steps is provided but will struggle to complete a pattern when provided with say steps 1, 3 and 5.

Allow students time to reflect after either completing their own patterns or one that was just completed on the board before asking for student explanations.

LM completes a pattern on the board and then selects a student to come up and explain the process involved in reaching the solution. Allow students to come up to the board and gesture to the different steps in the working to assist their explanation. Correct the student when incorrect terminology is used.

Students sit in a circle on the floor, LM uses large number cards to create a pattern providing students with the first few steps and then asking them to complete a few extras. Students work collaboratively to identify the rule. LM removes two steps from early on in the pattern and asks students to explain what numbers go in the missing spots using the rule that they just identified. Repeat the activity until students can complete and incomplete the pattern.

LM uses guided instruction to assist the class in completing an incomplete pattern on the board, encourage students to use prediction and guessing to find the missing sections.

Origin: LMs focus too heavily on growing patterns that are consecutive rather than using patterns that require students to find missing steps (Choi, Huh, LaRue, 2010).

The use of concrete materials appeared to assist many children (to) ascertain the missing steps in the pattern. A number, when completing the accompanying pen and paper worksheet, recreated the pictorial pattern with the tiles and then used the tiles to create the 5th and 10th step. They then drew a picture of their solution on the worksheet (Warren, 2005).

Students can often express the generalisations orally, but such descriptions often lack precision (Warren, 2005).

Fluency in the language of algebra (is) demonstrated by (the) confident use of its vocabulary and meanings as well as flexible operation upon its grammar rules (i.e. mathematical properties and conventions) (Panasuk & Beyranevand, 2010).

ACARA Links: The Number and Algebra strand has strong links to this challenge in that students will be using whole numbers, and all four operations when creating and solving their functions.

Year 6 and 7

Converting a visual pattern to a table of values so as to identify relationships from within the table (Choi, Huh, LaRue, 2010).

Example: Students may have a growing pattern where there are red counters and blue counters. Red increases at the rate of 3 each step and four at the rate of 5. Students may be able to model this using concrete materials but not know how to set up a table to represent the pattern and interpret it.

Explicit questioning to relate the position to the patterns visual components.

Explicit teaching on the different elements of a table, such as columns, rows, headings, values etc. Create a class poster for later reference when students are creating their own tables.

The LM provides students with a function and a set of values. The LM displays around the classroom a few different tables where the function and values have been incorrectly placed but only one that is correct. Allow students to select the poster which they believe is correct before asking students to provide explanations as to why they believed certain posters were incorrect.

Diagnostic Assessment: Students are to receive a homework activity where they are to identify a pattern within their home environment, one that is either present or one that they can create and are to record the values in a table. This table is to be used in the class for discussions to assist the LM in identifying areas of need in students learning.

Analyse poetry books to identify how the poetry is written and what pattern it follows. All data is to be recorded in a table that will be displayed around the classroom.

Students use Microsoft excel to record their values in a table and use the calculation functions included in excel to identify the function

Origin: Students are unsure as to how the pattern would fit into a table, such as which columns should be assigned which title ( Government of Newfoundland and Labrador, n.d.).

Patterns where the relationship between the pattern and position were explicit tend to assist students to identify and describe relationships that can be found when working with patterns.

(Warren, 2005).

Explicit questioning assists students in making connections between a particular position in a table to the visual components of a pattern (Warren, 2005).

Focusing on pattern generalisations before moving onto larger patterns is important due to teaching students the basics of working with tables, such as the correct terminology. This terminology can then be transferred to discussions involving larger patterns (Warren, 2005).

Tables of values assists students in constructing algebraic notation in a meaningful way (Booker et al. 2010).

Searching for patterns enables students to tie together new and prior knowledge, they also gain a greater conceptual understanding of the world of mathematics, and become better problem solvers

(NJDED, n.d.)

ACARA Links: The Money and Financial Mathematics content descriptor links here because the LM can present the patterns to student in the form of money related problems. The monetary values can than be transferred into a table.

Year 6 and 7

Students creating their own patterns (Choi, Huh, LaRue, 2010).

Example: Students may have learnt the skill of identifying rules within a LM provided pattern but have no understanding of how to create their own.

Guided class activity where students direct the LM on how to complete a pattern on the board, the LM corrects students on incorrect terminology usage.

Use concrete materials to create patterns, specific questioning to make explicit the relationship between the pattern and its position, and specific questioning that assists children to reach generalizations with regard to unknown positions (Warren, 2005).

Repeated opportunities for students to create their own patterns, these can be provided to their classmates (after being LM checked) to be completed.

Students use ICT learning objects to attempt to create their own patterns using a variety of different objects.

Diagnostic Assessment: Students are to collect materials from around the schoolyard / classroom to create a large pattern.

Origin: Students too often complete LM provided patterns rather than creating their own (Choi, Huh, LaRue, 2010).

Algebra mastery comes from lessons that are relevant to students, include real life situations and are multidisciplinary

(International Centre for Leadership in Education, 2007).

Orton and Orton (1999) described number patterns as a popular tool for developing childrens ability to express generality, which is argued to be one of four routes to algebra.

ACARA Links: The Money and Financial Mathematics content descriptor links here because students can use money to create their patterns. By using materials that students are familiar with, students will more readily participate and attempt the LM given problems.

Year 6 and 7

Understanding of the equals sign and how it is representative of quantitative sameness (Vance, 1998).

Example: Students may not understand that the equals sign signifies that the two different sides of an equation are equal, such as 2x6= 3x4.

To reinforce the concept that a number can be represented by many different expressions, learners are asked to name a given number in several different ways using two or more numbers and one or more operations. For example, 9 can be named as 4 + 3 + 2 or 2 x 5 1 (Vance, 1998).

In a related activity, equations are to be completed so that an expression has at least one operation on each side of the equals sign. For example, given 7 + 5 as one side of an equation, a student might write 7 + 5 = 2 X 6 or 7 + 5 = 10 + 5 3 (Vance, 1998).

As a class select a certain type of object where each one is of equal mass. The LM uses these objects to model a function where there is an equation on both sides of the equals sign. Students predict as to whether both sides of the equation are equal (using paper working). The LM uses the concrete objects and the scale to weigh up each side of the equation, providing students with a visual of equivalence. Repeat this activity with a variety of different equations, testing it with the scale each time.

This activity can also be completed using a variety of different objects, have students create the equations and test larger objects using a see-saw in place of a scale.

Origin: This misconception is reinforced early on when they are shown the vertical computational format for 3 + 2; the bar under the lower number is a signal to find an answer. It is also true that pressing 3 + 2 = on a calculator results in the standard form of the number. Therefore, children need experiences in which they see and write other types of number sentences, such as 5 = 3 + 2 and 3 + 2 = 4 + 1 (Vance, 1998).

An understanding of the mathematical language used in functions is necessary so as to enables students to organize and consolidate their mathematical thinking, to communicate with others; express mathematical ideas coherently and clearly to peers, teachers, and others; and use the language of mathematics as a precise means of mathematical expression

(Waters & Kelley, n.d.).

ACARA Links: The Number and Algebra strand has strong links to this challenge in that students will be using all four operations when exploring quantitative sameness.

Year 6 and 7

Variances in the standard number sentence of a + b = c (Choi, Huh & LaRue, 2010).

Example: Students may be so set in a particular number sentence that they dont understand that equations can vary. Such as 5+4=9 is one way, 7= 3+4 is another. The placement of the equals sign does not determine what is or isnt a correct number sentence.

Provide students with multiple opportunities to engage with a variety of different number sentences, explicit teaching on the fact that these variances are indeed all correct number sentences.

Explicit teaching on the different components of a function and their purpose. Make clear to students that their purpose does not change, regardless of where they are placed in an equation.

Diagnostic Assessment: Students write a brief paragraph on what they understand number sentences to be and another paragraph after a lesson on number sentences with any changes in their thinking. These are to be posted on the class wiki to document student learning.

Origin: Students who are used to seeing number sentences in the form a + b = c do not see 9 = 3 + 6 as a "proper number sentence"

(Choi, Huh & LaRue, 2010). Another possible origin is found in students misconception that alphabetical variables are representative of a specific number. For example a= 1, b=2, c= 3, d= 4 etc (Welder, 2012).

Students should be exposed to a variety of different number sentence problems. By allowing students to solve these problems, they will be able to build new mathematical knowledge through their work with problems; apply a wide variety of strategies to solve problems and adapt the strategies to new situations; and monitor and reflect on their mathematical thinking when solving problems. This practicing and reflecting will enable students to correct their misconception of what proper number sentences are (Waters & Kelley, n.d.).

ACARA Links: The Fractions and Decimals content descriptor links here in that students will be comparing different number sentences, some involving fractions and decimals.

Year 6 and 7

Bracket usage as a static signal telling students which operation to perform first in an algebraic equation (Gallardo, 1995).

Example: Students dont understand the order of operations. Students often ignore the brackets and dont understand that they are a type of operation not just a way to group number sets. Such as for 2x4 (7-2) -1, if students didnt use BODMAS, then they would get the incorrect answer.

Expose students to multiple equations which include brackets, assist students in completing these equations by changing teaching strategies from modelled to guided to individual work.

Explicit teaching of BODMAS, make clear the meaning of each letter and the purpose of each one in an equation.

Students create a small table stating the BODMAS rule that can be glued into their books to be referred to later.

Diagnostic Assessment: Students write written explanations explaining the process by which an equation needs to be completed.

Establish an activity where students are to write BODMAS next to each equation they need to complete and cross off each letter as it is considered / completed.

Explicit teaching of the bracket as the first order of operation, also its function as a mathematical symbol for multiplication of the value to the left of the bracket and the value/s within.

Origin: Students tend to solve such expressions based on how the items are listed, in a left-to-right fashion (in the case of English speaking students), consistent with their cultural tradition of reading and writing. Therefore, the rules underlying the order of operations can actually contradict students natural ways of thinking.

(Welder, 2012)

Success depends on students recognising that the order in which operations are carried out can affect the answers that you get. It is not always correct to perform calculations working from left-to-right (DEECD, 2010).

Students should be exposed to a wide variety of mathematical symbols used in algebra, such as brackets, so as to enable them to learn what these symbols represent and require the students to do when analysing this problem (Waters & Kelley, n.d.)

Students are unaware of the sequence of operations (BODMAS) and the purpose of brackets. This therefore results in students being unaware as to how to approach algebraic problems involving brackets in particular. (MacGregor & Stacey, 2003).

ACARA Links: The Number and Place Value content descriptor can be linked here because students will need to use all four operations when solving different equations, in particular using the BODMAS strategy.

Year 8 and 9

Working is commonly not shown by students and even though they may get the right answer they are often marked on their process

Full working also helps the students in reflecting on what part of the equation they made a mistake in and then reworking that area

At the beginning of teaching formulas provide students with boxes and lines for their working

Use a scaffolded approach so that you begin the teaching sequence with an expectation of full working

Model full working on the board using think alouds so that the teaching is explicit

Use an approach of modelled, guided and then individual work

During the guided approach use student direction and ask students for input into the formula e.g. the students think of the values of the variables or the situation of the formula

Bobis, Lowrie, Mulligan and Taplin (1999) support the view that mathematics must be student directed and cannot be entirely content driven or teacher centred if it is to be effective.

Origin: students do not see the benefit in showing their work but rather they see it as a waste of time. Especially when they understand the steps in their head and as a result find the correct answer (Allman, 2011).

It is important that all working is shown so that teachers can justify whether the correct method was used (Trimboli, 2011)

Year 8 and 9

Students often memorise formulas but do not understand them and hence when they have to apply them in a different situation they cant (Garlikov, 2000)

When teaching like and unlike terms make sure that the students create their own activity in which they will show what is and what isnt a like or unlike term

The students then swap their activity with a peer who will complete it and they will then correct it

Apply the formulas to the students current real life situation e.g. the height of all the students, the probability that a pencil in my case will be red etc can be worked out using algebraic formulas

Make sure that the formulas are taught in a meaningful context and are related to the students pre-existing understandings.

Students need to understand the importance of formulas rather than working them out in a left to right fashion (Welder, 2008). An example to teaching the importance of formulas and why we use them is to give two examples which need the students to follow the answers to achieve the answer e.g. 3 x 3 + 2 = 15 needs to have brackets e.g. 3 x (3 + 2) = 15 (Welder, 2008).

Memorising formulas originates from the practice of rote learning, which is less effective than schematic learning (Skemp, 1987). Schematic learning is more effective as the mathematical formulas and concepts fit into the students pre-existing understandings, and hence build upon their knowledge (Skemp, 1987).

Year 8 and 9

The concept of the left side must equal the right side has not been taught and therefore the students make a change only to one side of the equation (Garlikov, 2000)

Link to a visual diagram of a scale and use a physical example of a scale balancing

A kinesthetic example could be to use a see-saw in the playground and show how equations always need to be balanced

An origin for the challenge of inequality is to do with the teaching process. If the teacher has only focused on the formula and not the concept of balancing the equation then this will be reflected in the students work (Tipps, Johnson, & Kennedy, 2011).

Year 8 and 9

Maths has its own language and own vocabulary.

Therefore students need to know what the terms and definitions are, but they also must be able to give examples of these to demonstrate a practical understanding (Hutnick, 2004

Strategy / Activity: Students engage in a think pair share of algebraic terms and definitions

ICT: Students create posters and visuals to go with the definitions, terms and practical examples of each these will be displayed in the classroom. These can be made online at: http://www.glogster.com/

Peer Teaching: In groups students are given a couple of terms which they create definitions for, with aid from the text book and the internet. The students then create online posters and spend a couple of minutes explaining to the rest of the class what their definitions were and what are examples of each.

Teachers must also have an in-depth knowledge of all mathematical processes which they are teaching

Use a diagnostic test to see if students can match examples of terms to the correct terminology

A teacher cannot teach what they dont know, therefore they need to have a deep understanding of the processes that are far beyond the students (Bobis, Lowrie, Mulligan & Taplin, 1999).

This challenge originates from a lack of constructivist teaching, as the conceptual language is not understood before the concepts themselves are taught (Lester, 2007).

Displays can make a learning environment a cheerful and inviting place, but they can also enhance the self-esteem of students who help to create the learning environment through their work (Bobis, Lowrie, Mulligan & Taplin, 1999).

Students can be taught formulas and language from the text book but to understand mathematics and to develop their problem solving skills they need to work out non-routine problems (Cowan, 2006). Cowan (2006) further suggests that practical understanding comes from understanding the language of mathematics and from solving problems in relevant contexts.

Year 8 and 9

Often there is a misconception of equivalence and the equals sign is not seen as an expression Therefore students may say that the blank in this problem is 12 or 17:

8 + 4 = __ + 5

(Baroudi, 2006)

Solution: when using mathematical language try and use the term equals less but rather focus on each side of the equation e.g. 2+3 = 5 the left hand side is 2 plus 3

Solution: teachers need to provide more meaningful instruction

Solution: use a strip diagram where there the pictures are used for meaning not embellishment, the diagram is succinct and the page uncluttered (Beckmann, 2004)

The use of equations that require students to work out not just the number after the equal sign e.g. 6 + 6 = 8 + _

The equal sign misconception can also be related to the greater than and less than sign, to reinforce that like these signs it has a relational meaning

Use a diagnostic test to see which students need assistance with the misconception of an equal sign, for example the questions may be:

20a + 17a = __ - 16a

2(6a + 2a) = __ + 2a

The origin of this misconception stems from students only doing equations that result in one number after the equal sign, hence they see it as a means to one number rather than a relation to that whole side of the equation (Welder, 2008).

Some students are taught the meaning of the equal sign as the answer follows which results in common misconceptions that one side equals all of the other side (Foster, 2007). A strategy to solve this misconception is to use a scale and adding different sized and numbers of weights to each side, but so that it still remains equal. The middle of the scale is assumed to represent the equal sign. This helps to negate the equal sign misconception (Foster, 2007).

Year 8 and 9

Students lack a basic understanding of other mathematical concepts and so when they come to work out an algebraic equation with bracketing, they may not be able to if they do not understand multiplication e.g. positives and negatives, fractions, times tables (Garlikov, 2000)

Scaffold learning so the concepts are taught as they build on the students prior knowledge e.g. variables and expressions and then the substitution of variables and then using formulas etc (Philips & Strasser, 2011)

Make use of concrete items to teach students who have kinesthetic or visual learning styles

Strategy: use an expander, similar to the place value expander, but use it to show that between the variable CHG is a multiplication sign (C x H x G)

ICT: Use the interactive whiteboard to get students to rearrange formulas, collect like terms or substitute etc

Concrete materials assist all students in all year levels as long as they assist in teaching the content, (Bobis, Lowrie, Mulligan & Taplin, 1999). However concrete materials need to be taught with reflection, and other styles of teaching such as pen and paper, as well as digital means (Bobis, Lowrie, Mulligan & Taplin, 1999).

This challenge originates from the issue of not teaching to a childs level but from the curriculum. Even though a child may be in grade eight, their addition or multiplication may still be at a year six level. Hence a students prior knowledge always needs to be recognised and built upon (Lester, 2007).

Linear and non-linear relationships: students need to know the use of variables as symbols, the place of concrete materials and how to guess and check their work (ACARA, 2010)

Number and place value: students need a comprehensive understanding of multiplication, division, addition and subtraction. Students also need to be able to use these processes mentally as well as through written and digital mediums (ACARA, 2010)

Year 8 and 9

The ideas in algebra are abstract e.g. you cant touch or see them (Garlikov, 2000)

Use real life applications for problems or to solve problems that the students themselves may encounter in everyday life or after school e.g. The formula to encounter interest if the students would like to get a car loan

ICT: Use Google to search for real life car prices and loan terms

Use real life applications to work out the chance of winning the lottery etc so that students can see the place of algebra within a facet of society

When using visuals to add meaning for a problem use a strip diagram, so that the amounts for each part of the diagram are explicit.

Relate the students learning to money and finance as then it will be relevant for them put it into terms of casual jobs and pocket money, as well as savings

Students need to be taught mathematics using personal applications and real life experiences so that they become effective problem solvers (Bobis, Lowrie, Mulligan & Taplin, 1999)

Foster (2007) supports the idea that if students are taught abstract ideas without meaning there will be no understanding, hence this underlines that the origin of algebra being too abstract stems from teaching and learning that was not meaningful but was rather rote learning. Therefore the learning experiences taught should challenge misconceptions and build on the students prior knowledge (Foster, 2007). Hence the students need a lot of opportunities to work with tools to problem solve.

Money and Financial maths: students need to know the relation between profit, loss, the percentage of cost or sale prices and they need to be able to compare differences. Students also need to relate their algebraic thinking and mathematics to discount stores and retail store situations.