Issues Covered:

The issues that were covered this week related to geometry. Geometry involves the study of one-dimensional, two-dimensional and three-dimensional shape. Geometry is also the study of shape, space and measurement.

Understanding of mathematical concept

Geometry is where we realise that shapes and objects can be ordered into different areas such as 1D, 2D, and 3D shapes. There are many skills that are used in Geometry; we saw these in the lectures slides from week 9. These are shown below. Skill number 1 is visualising, skill 2 is communication, skill 3 is drawing and modelling, skill 4 is thinking and reasoning, the final skill, skill 5 is applying geometric concepts and knowledge.

Misconceptions:

A misconception students may have is that the rotation of a shape does not affect the original shape. For example, a square turned to make a diamond is still a square. To be able to help with this misconception we are able to get a group of shapes and rotate them so the students can visually see how the shape can be turned to make a shape without changing the actual shape. (Unknown, 2009)

Language Model

The language model that is pictured below has four main levels of language. Below in the table, it shows the language used in mathematics for patterning.

Language stages

Materials

Language

Recording

Students language

Rocks

Cans

sticks

Sides

Corners

No symbols, children drawing pictures only

Material Language

Attribute blocks

Bottles

Cans

Pattern blocks

Patterns

Sides

Faces

No symbols, children drawing pictures only

Mathematical language

Cubes

Pyramids

Cylinders

Faces

Area

Perimeter

Volume

No symbols, children drawing pictures may have some word stories

Symbolic Language

Pictures are usually used.

Teaching strategies:

There are a few teaching strategies we can do to help teach 3D shapes. These are by making them from paper, play-doh, clay etc. this is so the students are able to see how many faces they have. We are also able to sort shapes into 2D and 3D shapes; this will help students to be able to classify shapes into their dimensions.

Resources:

This first resource is one that was found in Scootle, it helps students to be able to name 3D shapes and discover the names of each of the shapes.

This second resource is one where students are able to classify shapes into 2D and 3D shapes.

ACARA:

  • - Foundation: Measurement and Geometry

o Shape

§ Sort, describe and name familiar two-dimensional shapes and three-dimensional objects in the environment (ACMMG009)

  • - Year 1: Measurement and Geometry

o Shape

§ Recognise and classify familiar two-dimensional shapes and three-dimensional objects using obvious features (ACMMG022)

  • - Year 2: Measurement and Geometry

o Shape

§ Describe and draw two-dimensional shapes, with and without digital technologies (ACMMG042)

§ Describe the features of three-dimensional objects (ACMMG043)

  • - Year 3: Measurement and Geometry

o Shape

§ Make models of three-dimensional objects and describe key features (ACMMG063)

o Geometric reasoning

§ Identify angles as measures of turn and compare angle sizes in everyday situations (ACMMG064)

  • - Year 4: Measurement and Geometry

o Shape

§ Compare the areas of regular and irregular shapes by informal means (ACMMG087)

§ Compare and describe two dimensional shapes that result from combining and splitting common shapes, with and without the use of digital technologies (ACMMG088)

o Geometric Reasoning

§ Compare angles and classify them as equal to, greater than, or less than, a right angle (ACMMG089)

  • - Year 5: Measurement and Geometry

o Shape

§ Connect three-dimensional objects with their nets and other two-dimensional representations (ACMMG111)

o Geometric Reasoning

§ Estimate, measure and compare angles using degrees. Construct angles using a protractor (ACMMG112)

  • - Year 6: Measurement and Geometry

o Shape

§ Construct simple prisms and pyramids (ACMMG140)

o Geometric Reasoning

§ Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles (ACMMG141)

(Australian Curriclum, Assessment and Reproting Authroity, 2015)

Textbook Synthesis:

This weeks reading focused on geometry. There were five main subjects that were focused on in the reading these were;

  1. Geometry — 2-dimensional shapes and 3-dimensional objects
  2. Location — position and movement of objects, maps, pathways and plans
  3. Transformation — translation (slide), rotation (turn) and reflection (flip)
  4. Symmetry of patterns, pictures and shapes
  5. Visualization and spatial reasoning. (Reys, et al., 2012)

For students to be able to work with 3D shapes they must be able to understand the concept of 2D shapes. When teaching 3D shapes it is usually easier for students to visualize the shape once they have made it. One of the first things students will focus on when working with 3D shapes is the number of sides or faces they have on it. Students will also learn about symmetry during geography, this is where students can cut up shapes and see if they are the same on each side. Again this is taught usually hands-on so students can physically see what happens.

REFERNCE LIST (WEEKS 5-9)

Australian Curriclum, Assessment and Reproting Authroity. (2015). Mathematics. Retrieved from Australian Curriculum: http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1

Reys, R., Rodges, A., Falle, J., Bennet, S., Fied, S., Smith, N., et al. (2012). Helping Children Learn Mathematics. Milton: Wiley.

Unknown. (2009). Misconceptions in Geometry. Retrieved from Geometry moduels: https://geometrymodule.wikispaces.com/file/view/Misconceptions.pdf

Victoria State Government. (2009). Measuring lenght. Retrieved from Education Victoria: https://www.eduweb.vic.gov.au/edulibrary/public/teachlearn/student/mathscontinuum/readmeaslength.pdf

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Issues Covered:

The issues covered this week were measurement. Measurement has many different counting units; Minutes, Days, hours, years, centimetres, metres, metres squared etc. Measurement is important as it is used in everyday life, these are things such as money, moving furniture, measuring when cooking etc. This week as there are so many units to cover in measurement I will be focusing on length.

Understanding of mathematical concept

To understand length we must know what to measure. For length, we must know how to measure using millimetres, centimetres, metres and kilometres. For students to be able to learn how to measure a length it is important for a student to have a direct comparison. For students to be able to understand the concept of length they must experiment with measuring different lengths such as a metre, a millimetre etc. this is so they have an understanding of how long something might be. Students will also need to learn the name of the units and their symbols such as millimetres = mm.

Misconceptions:

A misconception that students may have when measuring length with a ruler is to where to start. Some students may believe that they are to start at the end of the ruler. Some student may also start from 1 instead of 0 on the ruler. (Victoria State Government, 2009) It is important for us as educators when teaching students how to measure that we look at the equipment we have and look at where our starting point is. This way we are able to get students to measure correctly from the area we are supposed to measure.

Language Model

The language model that is pictured below has four main levels of language. Below in the table, it shows the language used in mathematics for Measuring Length.

Language stages

Materials

Language

Recording

Students language

Tall/er

Short/er

Longer

Smaller

Drawing pictures

Material Language

Toys

Hands

Counting numbers of non-standard unity

Drawing pictures or stating units they are using.

Mathematical language

Ruler

Trundle wheel

Centimetre cubes

Counting numbers in standard unit

Displaying answers in units

Symbolic Language

Rulers, tapes, speedometer

m, cm, mm, km

Using numbers and units

Teaching strategies:

A strategy to get students to think about measurement is by getting them to estimate. They would do this before measuring and measure it. Once they have measured it we would asked them how they measured it and how they got the answer. For example if a student said they got to a 4 on the ruler when measuring something we would then asked what does that mean? Or how do you know its four? From this we will push students into thinking about how they are doing it and why they would use the resource they used to do it.

Resources:

The first resource is a resource that students understand how to measure. It explains about having a straight stick to be able to measure.

http://www.pbslearningmedia.org/resource/619d53ff-1c4b-46bf-9082-0fa97a397576/treasure-map-peg-cat/

This second resource is one students can use when they have started looking at how to measure with a ruler, it asks students to measure the length of a line with a on screen ruler.

http://mathszone.webspace.virginmedia.com/mw/ruler/ruler_cm.swf

ACARA:

  • Foundation: Measurement and Geometry

o Using units of measurement

§ Use direct and indirect comparisons to decide which is longer, heavier or holds more, and explain reasoning in everyday language (ACMMG006)

§ Compare and order duration of events using everyday language of time (ACMMG007)

§ Connect days of the week to familiar events and actions (ACMMG008)

  • Year 1: Measurement and Geometry

o Using units of measurement

§ Measure and compare the lengths and capacities of pairs of objects using uniform informal units (ACMMG019)

§ Tell time to the half-hour (ACMMG020)

§ Describe duration using months, weeks, days and hours (ACMMG021)

  • Year 2: Measurement and Geometry

o Using units of measurement

§ Compare and order several shapes and objects based on length, area, volume and capacity using appropriate uniform informal units (ACMMG037)

§ Compare masses of objects using balance scales (ACMMG038)

§ Tell time to the quarter-hour, using the language of 'past' and 'to' (ACMMG039)

§ Name and order months and seasons (ACMMG040)

§ Use a calendar to identify the date and determine the number of days in each month (ACMMG041)

  • Year 3: Measurement and Geometry

o Using units of measurement

§ Measure, order and compare objects using familiar metric units of length, mass and capacity (ACMMG061)

§ Tell time to the minute and investigate the relationship between units of time (ACMMG062)

  • Year 4: Measurement and Geometry

o Using units of measurement

§ Use scaled instruments to measure and compare lengths, masses, capacities and temperatures (ACMMG084)

§ Compare objects using familiar metric units of area and volume (ACMMG290)

§ Convert between units of time (ACMMG085)

§ Use ‘am’ and ‘pm’ notation and solve simple time problems (ACMMG086)

  • Year 5: Measurement and Geometry

o Using units of measurement

§ Choose appropriate units of measurement for length, area, volume, capacity and mass (ACMMG108)

§ Calculate perimeter and area of rectangles using familiar metric units (ACMMG109)

§ Compare 12- and 24-hour time systems and convert between them (ACMMG110)

  • Year 6: Measurement and Geometry

o Using units of measurement

§ Connect decimal representations to the metric system (ACMMG135)

§ Convert between common metric units of length, mass and capacity (ACMMG136)

§ Solve problems involving the comparison of lengths and areas using appropriate units (ACMMG137)

§ Connect volume and capacity and their units of measurement (ACMMG138)

§ Interpret and use timetables (ACMMG139)

(Australian Curriclum, Assessment and Reproting Authroity, 2015)

Textbook Synthesis:

In this week’s textbook reading, chapter 17, it discussed everything with measurement. Measurement is one of the main topics used in student’s everyday lives. Children must measure frequently and real life problems, children must also develop estimation skills and measurement activities should encourage discussion of ideas that have come from doing these activities. The four measuring process can help educators plan learning experiences, these are;

  1. Identifying the attributes being measured
  2. Measuring with informal units
  3. Measuring with formal units
  4. Applying measurement to real life contexts.

Overall measurement is important and lots of hands-on activities are important to help students over come misconceptions and to have a good understanding on the measurement concepts. 

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Issues Covered:

The issues that were covered this week related early algebra. The main aspects that were reviewed this week were;

  • Patterns
  • Repeating and growing
  • Early algebra
  • Prealgebra

For this week, I will be covering patterning. This stood out, as I did not realise how patterning was a type of algebra. I didn’t realise for students to learn patterns they need to do lots of hands-on work. I also didn’t realise that patterning can also be sorting – so sorting objects into colours, thickness, shapes and sizes etc. This would be just to begin with until they begin patterns later on.

Understanding of mathematical concept

Patterns are an important part of mathematics as it can help children organise their world and understand mathematics. (Reys, et al., 2012). There are many types of patterns. Two are; repeating patterns and growing patterns. A repeating pattern is a pattern that is repeated over and over again. A growing pattern is a pattern that grows or gets bigger each time. An example of this is; A-B-AA-BB-AAA-BBB. We use patterns to help but conceptual understanding and build number patterns such as skip counting.

Misconceptions:

There are a few misconceptions children have when making patterns. One misconception is one where children may believe they found the functional relationship (usually in a growing pattern). This is because they have recognised two steps in a growing pattern and has applied it to all the other terms. To help with this misconception we should have hands-on activities for students to work out what was wrong with their original idea.

Language Model

The language model that is pictured below has four main levels of language. Below in the table, it shows the language used in mathematics for patterning.

Language stages

Materials

Language

Recording

Students language

Beads

Buttons

-

No symbols, children drawing pictures only

Material Language

Counters

Lego

How are the counters the same?

What order do you have them in?

No symbols, children drawing pictures only

Mathematical language

Unifix cubes

What attributes are there?

No symbols, children drawing pictures may have some word stories

Symbolic Language

Patterning does not use symbols or symbolic language.

Teaching strategies:

To be able to help students learn about patterning, it is important we have a hands-on perspective. One strategy for students to learn patterns in often taught in counting – skip counting. Children learn for example the skip two counting, which is a pattern of saying every second number. One strategy for helping students learn a pattern is by, having them sit in a girl/boy order or girl, girl, boy, boy order and stating they are sitting in a pattern.

Resources:

This resource here is one that will help explain what a pattern is to students – this is a video·

This next resource is one that helps students finish patterns in a fun way.

These resources are helpful in a classroom setting, they can both be used as a full class activity or one student can do on their own.

ACARA:

  • - Foundation: Number and algebra

o Patterns and algebra

§ Sort and classify familiar objects and explain the basis for these classifications. Copy, continue and create patterns with objects and drawings (ACMNA005)

  • - Year 1: Number and algebra

o Patterns and algebra

§ Investigate and describe number patterns formed by skip-counting and patterns with objects (ACMNA018)

- Year 2: Number and algebra

Patterns and algebra

§ Describe patterns with numbers and identify missing elements (ACMNA035)

(Australian Curriclum, Assessment and Reproting Authroity, 2015)

Textbook Synthesis:

In this week's reading, chapter 15, it spoke about algebraic thinking. The Australian curriculum’s number and algebra strand helps students in primary and early secondary recognise patterns which will then, in turn, help students understand the concepts of variables and functions. (Reys, et al., 2012). Patterns are important for students to learn as they can help the student organise their world, which will help them with mathematics. There are two types of patterns that students learn, the first is a repeating pattern, and this is where a pattern is consistently being repeated. The second pattern is a growing pattern; this is where a pattern gets bigger with more elements. We are able to help children develop algebraic concepts and the use of algebraic thinking by questioning, helping students model problems, pattern and relations. We can also help students by getting them to generalise and justify their thinking. (Reys, et al., 2012)


Refernce list will be with week 9.

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Issues Covered:

The main issues covered this week were everything to do with place value. Place value is important as it is one of the cornerstones of our number system. We know that number knowledge consists of formal and informal ideas. Formal is what we call "numeration" and place value whereas informal is what we call number sense. Numeration is what is taught in schools such as reading and writing, and interpreting and processing numbers. I found it interesting to know that a lot of people still don’t understand place value to the full extent.

Understanding of mathematical concept

Place value is so important as we need it for addition, subtraction, multiplication and division. When using place value it is in a base 10 thus meaning, “…that any number can be represented by 10 digits (0-9)” (Reys, et al., 2012). In place value every digit has a position, this position equals its value. For example the 4 in 406 is 4 hundred whereas the 6 in 406 is 6 ones. So when using place value the most ones a column is 9 then it would have to be traded for a set of 10 which would belong in the tens column.

Misconceptions:

Students often have the misconception that zero is nothing or and empty set. This is not always the case in place value. Often zero is a placeholder for digits as it could mean for example 406 which means there are 4 hundred and 6 of something not 4 and 6 of something. Some students may also see it as expanded notation – thus meaning they see it as 40+6 making it 46. Which again it is not it is the 4 hundred and 6. To overcome this we must explore and teach children the zero concepts. This can be hard, as students have usually been told the zero is nothing. In place value however we must explain that it holds the place of a number and does not mean nothing.

Language Model

The language model that is pictured below has four main levels of language. Below in the table it shows the language used in mathematics for Place value.

Language stages

Materials

Language

Recording

Students language

Beads

Buttons

Toys

Houses

Ones

Tens

Put back together

No symbols, children drawing pictures only

Material Language

Counters

Place value mats

Paddle pop sticks

How many?

Trade

Places

No symbols, children drawing pictures only

Mathematical language

Unifix cubes

Counters

Place value mats

Paddle pop sticks

Expand

Trade

House

Ones

Tens

Hundreds

No symbols, children drawing pictures may have some word stories

Symbolic Language

Equations

Written numbers

Equations

Written numbers

Place value mat diagrams with houses and numbers

Teaching strategies:

For student to be able to learn about place value they must know trading. This is usually done with a hands-on activity. This activity (in the video below) I have seen been done with a year 1 class while on my practicum. The students first started as a class to learn what to do each with their own place value mat. Once the students were able to successfully do it on their own they went off and played on their own or with a partner depending on the number or dice we had. Each student also had a 100’s chart. They were to cross off the numbers they had made.


please note the video below does not belong to me

Resources:

http://www.sheppardsoftware.com/mathgames/placevalue/scooterQuest.swf- this resource would be good for the upper years when they have startedtalking about tenths and hundredths.

http://www.ictgames.com/sharkNumbers/sharkNumbers_v5.html

This resource is for students to recognise the 10’s and 1’scubes to be able to add them up.

ACARA:

- Year 1: Number and algebra

o Number and place value

§ Countcollections to 100 by partitioning numbers using place value (ACMNA014)

- Year 2: Number and algebra

o Number and place value

§ Recognise,model, represent and order numbers to at least 1000 (ACMNA027)

§ Group,partition and rearrange collections up to 1000 in hundreds, tens and ones tofacilitate more efficient counting (ACMNA028)

- Year3 – Number and algebra

o Numberand place value

§ Recognise, model, represent andorder numbers to at least 10 000 (ACMNA052)

§ Applyplace value to partition, rearrange andregroup numbers to at least 10 000 to assist calculations and solve problems (ACMNA053)

- Year 4 – Number and algebra

o Number and place value

§ Recognise, represent and ordernumbers to at least tens of thousands (ACMNA072)

§ Applyplacevalue to partition, rearrange and regroup numbers to at least tens ofthousands to assist calculations and solve problems (ACMNA073)

(Australian Curriclum, Assessment and Reproting Authroity, 2015)

Textbook Synthesis:

Chapter 8:

In the reading from chapter 8, we were able to discover information about place value. Place value is critical to our understanding and making sense of numbers. There are two key ideas that number sense and place value rests on these are; 1. Explicit grouping and trading rules and 2. The position of the digit determines the number being represented. Our number system is the Hindu-Arabic number system which has 4 key characteristics which are;

o Place value – the position of the number represents the value

o Base of 10 – our system has 10 digits (0-9)

o Use of zero –represents the absences of something

o Additive property – numbers can be written in expanded notation and summed with respect of place value.

The last key point from the reading in chapter 8 was modeling grouped and ungrouped objects. I thought this was important as students need to know that no matter what they have they can be singles or groups of ten. I find this very important when teaching place value.(Reys, et al., 2012)

Chapter 10:

Within this chapter, it stated, “Whilst curriculum documents and professional organisations continue to recommend the use of calculators and computers in Australian schools (AAMT 1987, 1996), calculators tend to be underutilised as a teaching resource” (Reys, et al., 2012). Thus meaning that we should be using calculators in the classroom but using them as teaching resources, it should be used as using a calculator requires thinking and knowing how to use it. Estimation is also important for students. This is where teachers need to move students away from thinking about the exact answer mindset so they are able to use the power of estimation. Estimation will help students be able to see if their answer seems correct when doing a mental sum.(Reys, et al., 2012)

Chapter 14:

The number theory helps children link what they have learnt to other aspects of mathematics. One of the first theories children will learn is classifying numbers mainly odd and even numbers. It is important for children to learn 40% of children in year 4 can classify 2 or 3 digit numbers into odd or even. While children learn multiplication and division the students are learning about multiples and factors. This again is another number theory children learn. (Reys, et al., 2012)


References will be put with week 9

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Issues Covered:

The issues that were covered this week related to pre/earlynumber. The main aspects that were reviewed this week were;

  • · Patterning – algebra
  • · Early number and pre number concepts
  • · Subitising and
  • · Counting

For this week I will be covering counting. There are 5counting principles that children learn when learning to count. This stood out, as I did not know how to knowwhen a child can count. There are also three different stages of counting theseare counting on, counting back and skip counting. (Reys, et al., 2012).

Understanding ofmathematical concept

There are 5 counting principles that are covered whenlearning to count these are;

1. One to one correspondence

2. Stable order

3. Cardinal principle

4. Abstraction

5. Order irrelevance

These are important as if a child can do these 5 they areable to count. “Countingprocesses reflect various levels of sophistication, beginning with rotecounting and eventually leading to rapid skip counting forward and backward.” (Reys, et al., 2012). Thus meaning while students arelearning to count they are also learning skip counting forwards and backwards.

Misconceptions:

A commonmisconception children have is that numbers always start from one every timethey start counting. The teacher may geta child start at the middle counter and count to the end. Once the childreaches the end, the child is then asked to continue counting on from the firstcounter – the child may then believe that the number of the first counter isone.

Language Model

The language model that is pictured below has four mainlevels of language. Below in the table it shows the language used inmathematics for counting. .

Language stages

Materials

Language

Students language

Beads

Buttons

Numbers 1,2,3,4 etc

Material Language

Counters

Lego

More than

Less than

Mathematical language

Unifix cubes

100’s chart

What attributes are there?

Symbolic Language

There are none

There are none

Teaching strategies:

· Counting on – getting the children to count onfrom what they already have e.g 5 counters and another 4 are added “5,6,7,8,9”

· Counting back – to help with subtraction –children are counting backwards from a number

· Skip counting – counting by 2’s, 5’s, 10’s etcthis will help with multiplication and division later on.

Resources:

number trains:

The Gingerbread Man game - this can also be used for counting, Matching and Ordering:

ACARA:

  • - Foundation: Number and algebra

o Number and place value

§ Establish understanding of the language and processes of counting by naming numbers in sequences, initially to and from 20, moving from any starting point (ACMNA001)

§ Connect number names, numerals and quantities, including zero, initially up to 10 and then beyond (ACMNA002)

§ Compare, order and make correspondences between collections, initially to 20, and explain reasoning (ACMNA289)

  • - Year 1: Number and algebra

o Number and place value

§ Develop confidence with number sequences to and from 100 by ones from any starting point. Skip count by twos, fives and tens starting from zero (ACMNA012)

§ Recognise, model, read, write and order numbers to at least 100. Locate these numbers on a number line (ACMNA013)

(Australian Curriclum, Assessment and Reproting Authroity, 2015)

below is where to find the scootle resources and the elaborations for the foundation year curriculum link.

-

Textbook Synthesis:

The reading from Rey’s et al stated there were manydifferent areas of counting and number sense. These include;

  • Patterns
  • Skip counting
  • Sorting, matching and comparing
  • Counting by rote
  • One to one correspondence, groupingand cardinal order and
  • Writing numerals

The textbook states that to understand patterns we require problem-solving skills,which is an important part of mathematical learning. The book also stated thatSubitising is important as it saves time, it helps develop more sophisticatedcounting skills and helps the development of addition and subtraction skills. Countingprocesses also reflect differ levels of sophistication. It begins with rote countingand moves up towards skip counting both forwards and backwards. Over ourschooling and further into our lives counting is extended and refined. Startingwith pre number helps students gain the ability understand large numbers, placevalue and help with the development of problem solving skills to help withalgebra further on in their schooling. (Reys, et al., 2012).


References will be added in week 9

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Issues Covered:

The main issues covered this week related to division.Division is taught alongside multiplication. Division is separating a numberinto equal parts. There are two different ways to undertake division these arepartition and quotation. Partition is where we know the total number of groupsand want to find out how many are in each group, whereas, quotation is where weknow the total and the number of each group but we want to find out the numberof groups. From the textbook I found the most important thing was to helpstudents learn division while they are doing multiplication, as they are ableto link the facts back on each other in fact families. I also found thedivision strategy of subtractive algorithm very informative and a way I believechildren would be able to understand.

Understanding ofmathematical concept

“Division is a mathematical operation which can beinterpreted in several different ways; (i) grouping (quotation)- how manygroups there will be, (ii) sharing (partitioning) – how many will be in eachgroup, (iii) ratio – comparison between two quantities” (Klerk, 2007). When using these different ways of divisionwe are able to use quotation and partitioning in primary schools. When teachingchildren how to use these types of division hands on or real life aspectsshould be used. Quotation is important as we find out the number of groupswhere as partitioning we find out how many is in each group e.g. how manycookies can I share with my 4 friends if I have 8 cookies (84=2).

Language Model

The language model that is pictured below has four mainlevels of language. Below in the table it shows the language used in mathematicsfor division.

Language stages

Materials

Language

Recording

Students language

  • Animals
  • Cars
  • Chickens
  • Pegs
  • Books
  • Share
  • Sharing
  • Give away
  • How many can I give to each of my friends

No symbols, children drawing pictures only

Material Language

  • Paddle pop sticks
  • Counters
  • Unifix cubes
  • Place value mat
  • Share
  • Sharing
  • How many
  • Make out of.

No symbols, children drawing pictures only

Mathematical language

MABS

  • Divided by
  • Division
  • Equals

Recording in words

Symbolic Language

None should be needed

3 2=2

82=4


Teaching strategies:

please note video was filmed by myself.

Resources:

There are many resources used for Division. These include;

  • Counters
  • MABS
  • Paddle pop sticks
  • Place value chart
  • Unifix cubes
  • Multiplication mat – doubles asdivision mat

Most materials that you see are used for many of thedifferent conceptions that children need to learn.

ACARA:

-     Year2: Number and algebra

o Numberand place value

§ Recognise and represent division as groupinginto equal sets and solve simple problems using these representations (ACMNA032)

-     Year3: Number and algebra

o Numberand place value

§ Recall multiplicationfacts of two, three, five and ten and related divisionfacts (ACMNA056)

-     Year4: Number and algebra

o Numberand place value

§ Recall multiplication factsup to 10 × 10 and related division facts (ACMNA075)

§ Develop efficient mental and written strategiesand use appropriate digital technologies for multiplication andfor division where there is no remainder (ACMNA076)

o Patternsand algebra

§ Solve word problems by using numbersentences involving multiplicationor division where there is no remainder (ACMNA082)

-     Year5: Number and algebra

o Numberand place value

§ Solve problems involving division by a one digitnumber,including those that result in a remainder (ACMNA101)

o Patternsand algebra

§ Find unknown quantities in numbersentences involving multiplication anddivision and identify equivalent numbersentences involving multiplication anddivision (ACMNA121) 

-     Year6: Number and algebra

o Fractionsand decimals

§ Multiplydecimals by whole numbers and perform divisions by non-zero whole numbers wherethe results are terminating decimals, with and without digital technologies (ACMNA129)

§ Multiplyand divide decimals by powers of 10 (ACMNA130)

(Australian Curriclum, Assessment and Reproting Authroity, 2015)

Textbook Synthesis:

“Division is the inverse of multiplication; that is, in adivision problem you are seeking an unknown factor when the product and someother factor are known.” (Reys, et al., 2012). Division isnot usually learnt on its own, division associates with multiplication aschildren start to realise that its multiplication facts but in reverse e.g. 486=8where as 8x6=48. Using real world problems is the best way to approachdivision, as children are able to relate to them, such as sharing lollies withtheir families. One of the most common strategies to help students learndivision is skip counting, like what was seen with addition/multiplication butin reverse. The other strategy that I thought stood out was using a subtractivealgorithm, where a child can subtract any multiple of the divisor at each step.This can be seen on page 265 in Reys, et al,. Overall linking subtraction andmultiplication to division will help students comprehend the concept.

Reference list (weeks 1-4)

Australian Curriculum, Assessment and ReportingAuthority. (2015). Mathematics. Retrieved from Australian Curriculum:http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1

Klerk, J. (2007). IllustratedMaths Dictonary (4th Edition ed.). Melbourne, VIC, Australia: Pearson.

Reys, R., Rodges,A., Falle, J., Bennet, S., Fied, S., Smith, N., et al. (2012). HelpingChildren Learn Mathematics. Milton: Wiley.

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Issues Covered:

The issues covered this week were in relation tomultiplication. Multiplication is taught after students learn addition, this isbecause multiplication is essentially repeated addition or addition of equalparts. There are four ways to view multiplication and these are; array models,set models, measurement and length models and the combination model. The aspectI found the most interesting about multiplication was the order it is taughtin. I originally thought that zero and ones tables would be taught first thentwos, fives and tens. Once it was explained that it is more difficult to teachthat way due to addition it made sense of why you would start with 5’s and10’s. I found this week informative and using the different models will helpwith my future teaching of multiplication.

Understanding ofmathematical concept

Multiplication is repeated addition or to multiply is to“carry out the process of repeated addition or multiplication” (Klerk, 2007).Multiplication is seen in 4 different ways but there is only one type ofmultiplication unlike subtraction. It is important for students to startlearning multiplication that they are confident with the addition concept. Thisis because when teaching students multiplication we begin with the set model.E.G. there are 3 worms in each of the 5 apples, how many worms are there alltogether? (Seepicture below) some students will try to count these out, where as we want themto use repeated addition, 3+3+3+3+3 = 15. The other ways to see multiplicationis by the array model, measurement and length and the combination models.

Language Model

The language model that is pictured below has four mainlevels of language. Below in the table it shows the language used inmathematics for multiplication.

Language stages

Materials

Language

Recording

Students language

  • Animals
  • Cars
  • Chickens
  • Pegs
  • Stacks of
  • Sets of
  • Bags of
  • In each row

No symbols, children drawing pictures only

Material Language

  • Paddle pop sticks
  • Counters
  • Hundreds/tens/ones table
  • How many?
  • Total of?

No symbols, children drawing pictures only

Mathematical language

MABS

  • Multiply
  • Multiplication
  • Times
  • Multiply by
  • Equals

No symbols, children drawing pictures may have some word stories

Symbolic Language

None should be needed

3x2=6

8x2=16

Teaching Strategies:

please note video was created by myself.

Resources:

There are manyresources used for multiplication. These include;

  • - Counters
  • - MABS
  • Paddlepop sticks
  • Multiplicationtables

Mostmaterials that you see are used for many of the different conceptions thatchildren need to learn.

ACARA:

-     Year2: Number and algebra

o Numberand place value

§ Recognise and represent multiplicationas repeated addition, groups and arrays (ACMNA031)

-      Year3: Number and algebra

o Numberand place value

§ Recall multiplicationfacts of two, three, five and ten and related divisionfacts (ACMNA056)

§ Represent and solve problems involving multiplication usingefficient mental and written strategies and appropriate digital technologies(ACMNA057)

-     Year4: Number and algebra

o Numberand place value

§ Recall multiplication factsup to 10 × 10 and related division facts (ACMNA075)

§ Develop efficient mental and written strategiesand use appropriate digital technologies for multiplication andfor division where there is no remainder (ACMNA076)

o Patternsand algebra

§ Solve word problems by using numbersentences involving multiplicationor division where there is no remainder (ACMNA082)

-     Year5: Number and algebra

o Numberand place value

§ Solve problems involving multiplication oflarge numbers by one- or two-digit numbers using efficient mental, writtenstrategies and appropriate digital technologies (ACMNA100)

o Patternsand algebra

§ Explore and describe numberpatterns resulting from performing multiplication (ACMNA081)

§ Find unknown quantities in numbersentences involving multiplication anddivision and identify equivalent numbersentences involving multiplication anddivision (ACMNA121)

-     Year6: Number and algebra

o Fractionsand decimals

§ Multiplydecimals by whole numbers and perform divisions by non-zero whole numbers wherethe results are terminating decimals, with and without digital technologies (ACMNA129)

§ Multiplyand divide decimals by powers of 10 (ACMNA130)

(Australian Curriclum, Assessment and Reproting Authroity, 2015)

Textbook Synthesis:

The readingfrom Rey’s et al. a statement that stood out to me was “The basicmultiplication should not be given to children in the form of a table or chartof facts until they have been meaningfully introduced.” (Reys, et al., 2012). This is sothe facts are developed through problem situations, experiences with hands onequipment, pictures and various thinking stages. Repeated addition is aneffective way to learn multiplication for facts under 5, another great strategyas shown above in the video is splitting the product. This is where studentscan simplify the multiplication. When teaching children multiplication oflarger numbers it helps children if we are using place value charts. This is sothey don’t ignore any zeros in the chart e.g. a child might have the question306x9= (their answer) 324, where as when we have it in a place value chart theyare able to see that it would be (9x300)+(9x6) which equals 2700+54 = 2754.Thus using expanded notation will help students understand the correct procedure.

*all references are with week 4*

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Issues Covered:

The issues covered this week related to subtraction and howit is taught in a school setting. The main aspects that I reviewed from this week werethe three different types of subtraction and how it relates to addition. Thethree types of subtraction stood out to me, as I had never really heard aboutthem before. I thought subtraction was just subtraction and didn’t realise howthe wording would make a difference into what type of subtraction it was. Iknew that subtraction related to addition but I didn’t realise we could use thesame strategies such as the use 10 strategy or addition mats but in reverse.

Understanding ofmathematical concept

Subtraction is when you know the total and one part of thetotal. Thus meaning you have to find out the missing part (second part to thetotal). Subtraction has three different types of subtraction, takeaway,difference/comparison and missing addend. There are many strategies that areused for subtracting – most of these are similar to addition. Doubles is astrategy that is used when the students know doubling with addition.Subtraction can only really be taught once students start to understand theaddition concept.

Language Model

The language model that is pictured below has four mainlevels of language. Below in the table it shows the language used inmathematics for subtraction.

Language stages

Materials

Language

Recording

Students language

  • Animals
  • Cars
  • Chickens
  • Pegs
  • Went away
  • Buzzed off
  • Taken away
  • Eaten

No symbols, children drawing pictures only

Material Language

  • Paddle pop sticks
  • Counters
  • How many more?
  • How many less than

No symbols, children drawing pictures only

Mathematical language

MABS

  • Subtract
  • Equals

No symbols, children drawing pictures may have some word stories

Symbolic Language

None should be needed

8-4=4

8-4=4


Teaching strategies:

*please note video was created by myself*

Resources:

There are many resources used for subtraction. Many of thesewere seen with addition as well, they include;

  • Counters
  • MABS
  • Paddle pop sticks
  • Addition mat (used as a subtraction mat)
  • Subtraction stories

Most materials and resources that are used for subtractionare also used in other concepts throughout the schooling year.

ACARA:

-       Year 1: Number and algebra

o Number and place value

§ Represent and solve simple addition and subtraction problems using arange of strategies including counting on, partitioning and rearranging parts (ACMNA015)

-      Year 2: Number andalgebra

o Number and place value

§ Explore the connection between addition and subtraction (ACMNA029)

§ Solve simple addition and subtraction problems using a range ofefficient mental and written strategies (ACMNA030)

-      Year 3: Number andalgebra

o Number and place value

§ Recognise and explain the connection between addition and subtraction (ACMNA054)

§ Recall addition facts for single-digit numbers and related subtractionfacts to develop increasingly efficient mental strategies for computation (ACMNA055)

o Patterns and algebra

§ Describe, continue, and create number patterns resulting fromperforming addition or subtraction (ACMNA060)

-      Year 4: Number andalgebra

o Patterns and algebra

§ Find unknown quantities in number sentences involvingaddition and subtraction and identify equivalent number sentences involvingaddition and subtraction (ACMNA083)

-      Year 5: Number andalgebra

o Fractions and decimals

§ Investigate strategies to solve problems involving addition andsubtraction of fractions with the same denominator (ACMNA103)

o Patterns and algebra

§ Describe, continue andcreate patterns with fractions, decimals and whole numbers resulting fromaddition and subtraction (ACMNA107)

-     Year 6: Number andalgebra

o Fractions and decimals

§ Solve problems involvingaddition and subtraction of fractions with the same or related denominators (ACMNA126)

(Australian Curriculum, Assessment and Reporting Authority, 2015)


Textbook Synthesis:

The reading from Rey’s et al explained about concrete andabstract learning. Abstract for students might not only be y + x =but just symbols may seem abstract to students in the lower years.Students should begin with hands on learning or concrete materials to helpunderstand a concept. The reading also suggested that to help students learn weshould have an encouraging classroom that supports all learners. In chapter 11it stated that “children need to become comfortable with the idea that 37 canbe seen as 3 tens and 7 ones… not just as no tens and 37 ones.” (Reys, et al., 2012). Because ofthis students will be able to develop the flexibility to be able to understandstrategies needed in subtraction.

*reference list given at end of week 4*

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Issues Covered:

The issues that were covered this week were related toaddition. The main points that I received from this weeks content included;language models, what addition is and breaking down addition. The languagemodels; student language, material language, mathematical language and symboliclanguage, helped myself as a future educator be more familiar with words thestudents will be using and will know, as well as what types of recording theywill be using. Breaking down addition was something I didn’t know how to do andfrom the readings, lecture and tutorial I have now understood how to teachaddition to students, especially in younger years.


Understanding ofmathematical concept

Addition is the joining two or more objects or symbols togetherto make a total. The concept to understand in addition is that we are joiningobjects together to make a total. The skills that are used in addition is howdo we join objects or symbols to make a total and how do we add two digits?Some strategies covered in the reading by Reys et al, included adding one ormore, doubling or near doubling, counting on, combination 10 (rainbow facts) (Reys, et al., 2012). Thesestrategies help students by breaking down the addition to something they feelmore comfortable with. Knowing these strategies can help later on with otherconcepts.


Language Model

The language model that is pictured below has four mainlevels of language. Below in the table it shows the language used inmathematics for addition.

Language stages

Materials

Language

Recording

Students language

- Animals

- Cars

- Chickens

- Pegs

- Buttons

- Came

- Arrived

- All together

- Put with

No symbols, drawing pictures only

Material Language

- Paddle pop sticks

- MAB’s

- Counters

- Counters

- Cubes/sticks

No symbols, drawing pictures only

Mathematical language

MAB’s

- Equals

- Add

No symbols, drawing pictures and could write word stories

Symbolic Language

None should be used

3+4=7

8+2=10


Teaching strategies:

In the photos below we are able to see a counting onstrategy. It can be used when adding one or two to the larger number. (Reys, et al., 2012). Students should count on using the largeraddend to make it easier. Some students will start from the lower addend dependingon the way the numbers are arranged. This teaching strategy will help thestudents learn to use the larger addend first.

Resources

Someresources that are used for addition include:

  •   Counters
  •  MABS
  •  Additionmats
  • Rainbowfact sheets

ACARA:

-      Foundation:Number and algebra

o  Numberand place value

§ Represent practical situations to model additionand sharing (ACMNA004)

-      Year1: Number and algebra

o  Numberand place value

§  Representand solve simple addition and subtraction problems using a range of strategiesincluding counting on, partitioning and rearranging parts (ACMNA015)

-      Year2: Number and algebra

o  Numberand place value

§  Explorethe connection between addition and subtraction (ACMNA029)

§  Solvesimple addition and subtraction problems using a range of efficient mental andwritten strategies (ACMNA030)

-      Year3: Number and algebra

o  Numberand place value

§  Recogniseand explain the connection between addition and subtraction (ACMNA054)

§  Recalladdition facts for single-digit numbers and related subtraction facts todevelop increasingly efficient mental strategies for computation (ACMNA055)

-      Year4: Number and algebra

o  Patternsand algebra

§  Findunknown quantities in numbersentences involving addition and subtraction and identify equivalent numbersentences involving addition and subtraction (ACMNA083)

-      Year5: Number and algebra

o  Fractionsand decimals

§ Investigate strategies to solve problemsinvolving addition and subtraction of fractions with the same denominator (ACMNA103)

-      Year6: Number and algebra

o  Fractionsand decimals

§  Solveproblems involving addition and subtraction of fractions with the same orrelated denominators (ACMNA126)

(Australian Curriclum, Assessment and Reproting Authroity, 2015)    

Textbook Synthesis:

The important aspects from the textbook readings (chapter1, 9 and 11) were that when learning addition children need to be taught itusing experiences they have had personally. In chapter 9 it states ‘Early on,children should be manipulating materials as they record answers’ (Reys, et al., 2012). To help children learn theaddition concept there is a sequence that is appropriate; concrete learning orlearning hands on, second is pictorial learning or learning with pictures andthirdly abstract learning or using symbols to illustrate an operation. Therewere many different thinking strategies mentioned in the readings, which arelisted in the understanding ofmathematical concept above.  Fromthis week’s readings the main points I learnt were to use materials and to makethe mathematical questions relate to the children or to something they recognise.

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