Preliminary Mathematics Stage 3

This course is organised into four sequential stages. Each stage includes a learning focus statement and examples of student learning. Each stage is designed to accommodate a learner's level of readiness and reflects their stage of development. It may be likely that a learner will engage in only one stage of learning in their study of this course.
The learning program focuses on ideas, skills, techniques and processes. It includes engagement in concepts that emerge from a range of starting points and stimuli. Learners explore experiences, ideas, feelings and understandings through making and responding.

This document represents Stage 3 of the course Preliminary Mathematics.

This course has been accredited under Section 55ZI of the Office of Tasmanian Assessment, Standards and Certification Act 2003 - Other education. ‘Other education’ is defined by the Act as including ‘education specifically designed for a person with a disability’. 

Section 55ZM of the Act requires that providers of this course issue qualifications to learners who achieve a Preliminary Achievement (PA) award or higher at any stage or size value. For further information and examples, see the Supporting Documents section of this course.

In addition to provider-issued qualifications, course providers may report learner results to the Office of TASC in late November each year using prescribed methods. Reported results will be used by TASC for inclusion on students’ Qualification Certificates (QCs) and may form part of the evidence base for a Tasmanian Certificate of Educational Achievement (TCEA) for eligible learners.

Achievement in courses Preliminary to Level 1 does not contribute to the Tasmanian Certificate of Education (TCE). This allows for greater flexibility in regard to learner movement between stages and their learning time (reflected in variable size values for this course) than are available in Level 1 and higher senior secondary courses.

The Preliminary to Level 1 suite of courses are designed for learners whose circumstances significantly impact their learning – in an ongoing manner or for extended periods of time. 

These learners include:

• learners with disability as defined by the Disability Discrimination Act (DDA) 1992, whose disability requires educational adjustments to support access and participation in learning
• learners whose circumstances have significantly impacted their ability to attend school for extended periods of time or engage in courses with higher levels of complexity
• learners who require significant support for learning English as an additional language or dialect.

CONTENT STRANDS
The content strands describe what is to be taught and learnt. They are outlined below.

Number and algebra
Within this strand, learners will engage in activities designed to develop their understanding of number and place value, react to situations involving money and identify, explore and predict patterns. They will correspond quantities with names and numerals and use this information to perform simple addition and subtraction. They will explore sharing quantities through partitioning. They will identify, relate and extend patterns and attributes in number series such as even numbers.

Measurement and geometry
Within this strand, learners will engage in activities designed to develop their understanding of the physical attributes of the world around them. They will define, sort and classify shapes and measure attributes including length, weight, temperature, time and capacity. They will describe, change and interpret location, position and movement of objects and show an awareness of time, both in intervals and the concept of time passing.

Statistics and probability
Within this strand, learners will engage in activities designed to develop their understanding of statistical information. They will experience data collection through observations, surveys and through the use of measurement instruments. They will explore ways that data can be represented using tables and charts. They will answer closed questions to describe data e.g. which response was the most popular?

PROFICIENCY STRANDS
The proficiency strands below describe how content is explored or developed and the actions in which learners can engage when learning and using the content.

Understanding
Understanding involves making connections between related concepts and progressively applying the familiar to develop new ideas. This includes connecting names, numerals and quantities, partitioning and combining numbers flexibly and identifying the relationship between addition and subtraction. It involves identifying, describing and manipulating attributes of shapes, using appropriate language to communicate properties of time and interpreting simple data sets.

Fluency
Fluency includes readily counting numbers in sequence, recalling addition and multiplication number facts and using estimation to check the reasonableness of answers to calculations. It includes using instruments to measure accurately and selecting appropriate units of measurement to order, compare and calculate attributes. It supports identifying and describing outcomes of chance experiments, interpreting maps and communicating directions and position.

Problem-solving
Problem-solving includes using materials to model authentic problems and using number sentences that represent problem solutions, using familiar counting sequences to solve unfamiliar problems and discussing the reasonableness of the answer. It involves comparison of large numbers, currency or time durations and using properties of numbers to find and extend patterns. It involves the ability to formulate and solve authentic problems through modelling and pattern.

Reasoning
Reasoning includes explaining comparison of quantities through direct and indirect comparison and through the use of informal and formal units. It includes explaining and justifying patterns, representations of data and results to problems. It involves using known facts to derive strategies for unfamiliar calculations and posing appropriate questions for data investigations and interpreting data sets.

This Preliminary to Level 1 course is organised using the three interrelated content strands from the Australian Curriculum: Mathematics and structured so that teachers can recognise how the four proficiency standards from the Australian Curriculum: Mathematics can be applied to assist content exploration and skill development.

Mathematics is the study of order, relation and pattern. From its origins in counting and measuring it has evolved in highly sophisticated and elegant ways to become the language now used to describe much of the modern world. Mathematics is also concerned with collecting, analysing, modelling and interpreting data in order to investigate and understand real-world phenomena and solve problems in context. Mathematics provides a framework for thinking and a means of communication that is powerful, logical, concise and precise. It impacts upon the daily life of people everywhere and helps them to understand the world in which they live and work.

Preliminary Mathematics caters for learners who require flexible and individualised programs. Learners will experience a range of learning activities in familiar contexts that will assist them to attend to and explore the world around them with as much independence as possible. Learning experiences in this course are designed to move learners from an emergent level of responding to a level where the response indicates intention.Preliminary Mathematics aims to offer better outcomes in all aspects of learners’ lives, and assist learners’ access to achievable pathways to further education, training and/or employment outcomes.

On successful completion of this course, learners will be able to:

1. identify whole numbers and apply fundamental number skills to everyday mathematical situations
2. use units of measurement
3. use spatial awareness to describe position, location and shape
4. collect, represent and interpret statistical information.

This course focuses on progressing learners from a pre-intentional to intentional state. It encourages learners to develop their independence as they explore, participate and engage in the world around them. Expectations for learner achievement are set at four stages which are not associated with any set age or year level that links chronological age to developmental progress. As learners progress through these stages the level of coactive support required decreases as they proceed towards becoming independent learners.

This document is Stage 3 in four Stages of the course Preliminary Mathematics.

Stage 1 
At this stage learners experience a range of learning activities that will assist them to attend to and explore the world around them with as much independence as possible. Experiences are designed to move learners from a pre-intentional level of responding to a level where the response indicates beginning intention. Learners will need high levels of coactive support and focused attention from the teacher to help them initiate and refine their responses. Learners demonstrate some awareness and recognition of familiar people and routine activities.

Stage 2
Learners at this stage become less reliant on high levels of coactive support and become more reliant on verbal prompts and gestures to facilitate their learning. They begin to explore their world independently and engage in simple cause and effect play activities. Learners are able to focus on structured learning activities for short periods of time. They respond to familiar people and events and begin to use ‘yes/no’ responses. 

Stage 3
Learners at this stage are less dependent on coactive support and respond more consistently to prompts and simple clear directions from the teacher to support them in their learning. They are displaying the first signs of independence and becoming more peer focused. Learners participate in structured learning activities with others and they begin to use pictures, photos and objects to communicate personal interests and experiences. They start to use and link some familiar words and images to form a meaningful communication.

Stage 4
With teacher support and curriculum scaffolding, learners at this stage participate cooperatively in group learning activities. They express their feelings, needs and choices in increasingly appropriate ways and combine and sequence key words and images to communicate personal interests and to recount significant experiences. They indicate beginning understanding of social rules and expectations and are beginning to reflect on their own behaviour.

Learners can enter the course at any of the four stages depending upon their level of readiness or ability. Providers will determine the entry point for learners based on each learner’s level of readiness.

All learners have a right to an education that meets their needs through full participation and engagement with learning. Providers enhance the quality of all students’ learning through responsive and planned learning programs that optimise achievement using inclusive practices.

Learners will have access to appropriate resources such as professional support staff (or adult) assistance, and equipment such as assistive technology or modified facilities, as required. The Disability Standards for Education Act 2005 outline the obligations that educational providers must meet in supporting the needs of learners with disability through reasonable adjustments.

Adjustments include a range of supports and accommodations including but not limited to:

• supportive learning environments 
• tailored delivery models 
• visual and organisational supports 
• multimodal learning and assessment opportunities 
• varying levels of prompting and adult assistance. 

‘Adjustment’ does not mean the lowering or removal of standards but rather it means allowing learners appropriate means to demonstrate their learning and achievement against standards.

In the instance of online, distance or flexible delivery, adjustments may be made in the delivery of the courses by providers to suit the individual student’s learning environment. This may involve differentiation to reflect the resources readily available to the learner within their home or community.

This course is preparatory to Everyday Maths Level 1.

This course has a complexity Preliminary to level 1.

Each stage of the course has a size value of 5 or 10 or 15 depending upon learners needs. A learner will undertake a minimum of 50 hours study in this course, through to a maximum of 150 hours dependent on learner needs.

There are many possible ways of combining the Stages of learning and the size value of 5 or 10 or 15 depending upon learners needs.
For example:

• A learner might study Preliminary to Level 1, Stage 3 of the course and complete it in 50 hours. This learner might then study Preliminary to Level 1, Stage 4 of the course and require 100 hours to achieve this. After successful completion, this learner might then undertake Everyday Maths Level 1 in the following year.
•  In one year a learner might study Preliminary to Level 1, Stage 2 of the course and complete it in 50 hours. This learner might then also study Preliminary to Level 1, Stage 3 of the course and require 100 hours to achieve this.
• A learner may study Preliminary to Level 1 Stage 1 in 150 hours.

This course is organised into four sequential stages. Learners can enter the course at any of the four stages depending upon their level of readiness or ability. Providers will determine the entry point for learners. 

This document is Stage 3 of the four stages in this course.

The sequence of delivery for each stage is at the discretion of the provider. A personalised learning program will be developed for each learner, dependent on their needs. Not all learners will achieve all stages of the course content.
While there is no prescribed order of delivery within each stage, the course material from each content strand must be completed and assessed against criterion 1 (number and algebra), criteria 2-3 (measurement and geometry) and criterion 4 (statistics and probability). To support effective learning it is recommended that providers design a program of study that combines/concurrently delivers the content in criterion 1 alongside the content in criteria 2 – 4. While not all proficiency strands apply to every part of the course content, they provide a meaningful basis for the development of concepts within mathematics and as such indicate the breadth of mathematical actions that teachers can emphasise to support learning of the prescribed content.

STAGE 3: (50, 100 AND 150 HOURS)

This stage has a size value of 5, 10 or 15.

In Stage 3, learners engage, explore and respond to experiences through communication with known adults, teachers and peers.
Within number and algebra, learners will recognise, match, add and count quantities up to ten, identify regular configurations of quantities through  subitisation (such as those occurring on dice, card and dominos), and describe difference between two quantities in various ways including through the introduction of the formal symbol for subtraction. They will sort coins and notes into groups and match coins with 2-Dimensional templates.

Learners will:

• play games involving dice, playing cards and dominoes that require recognition of quantities and counting both forwards and backwards (such as snap, snakes and ladders)
•  participate in songs, rhymes, games and stories that require counting forwards and backwards
• explore differences in quantities up to 5 through comparison using appropriate language of ‘more’, ‘less’ or ‘same’
• model situations of ‘taking away’ and describe the process in various way
• indicate (without counting) the larger of two groups when presented with significantly different number sets of objects
• using one-to-one correspondence to distribute materials evenly
• identify and group notes or coins based on a set criteria
• match coins with 2-Dimensional templates in ‘exchange of goods for cash’ simulations to purchase items in role play situations.

Within measurement and geometry, learners sort and classify a range of common two-dimensional shapes. They use an increasing range of terms to describe shapes and using malleable materials explore the attributes of 3-Dimensional objects and how they can be constructed from 2-Dimensional shapes. Learners respond to specific instructions relating to the position and location of self and objects and measure physical attributes including length, weight, capacity, time and temperature. Through participation in daily routines, learners will begin to identify days of the week and react to environmental time cues. 

Learners will:

• visualise and describe 2-Dimensional shapes not in sight or identify 2-Dimensional shapes from a series of descriptive statements
• create copies of 3-Dimensional objects using malleable material
• select equipment to conduct measurements to perform activities such as simple science experiments, design tasks, cooking or planning a garden plot layout.
• participate in stories, games and songs involving naming the days of the week in sequence, months of the year and season
• recall in order events that have occurred in a day or steps in a task using language including ‘before’, ‘after’, and ‘next’
• associate activities with nominal amounts of time (short, long etc.)
• measure temperature changes during the day or between days (once a week) and track how temperature changes over time (relate to time of day or year)
• participate in activities to identify and sequence days of the week in order (e.g. songs, chants, stories)
• give and follow instructions to locate an object not in sight or move to a location that is out of sight
• identify local or school landmarks in photographs. 

Within statistics and probability, learners will experience and participate in simple data collection through observation of simple characteristics (such as colour, gender, size) and/or through survey using closed questions. They will follow instructions to organise data into groups/categories and create data displays such as a pictorial frequency table. They will experience the simple language of chance.

Learners will:

• experience the collection, sorting and displaying of information and data 
• associate objects with familiar events in practical situations (e.g. recognising connection between own bag and lunchtime, paint and brushes with art class)
• experiencing people's choices being organised into a simple display
• react to organisation of shapes according to the number of sides
• experience and may respond during conversation about the likelihood of relevant events (e.g. ‘I think that shape will fit there’, ‘It might be a baby boy’)
• experience adults making meaning from statistical information (e.g. it looks like it will rain tomorrow).
• sort data responses into groups (e.g. sort photos of students according to information obtained from a survey such as number of siblings, number of pets, town they live in, how they get to school).

Criterion-based assessment is a form of outcomes assessment that identifies the extent of learner achievement at an appropriate end-point of study. Although assessment – as part of the learning program – is continuous, much of it is formative, and is done to help learners identify what they need to do to attain the maximum benefit from their study of the course. Therefore, assessment for summative reporting to TASC will focus on what both teacher and learner understand to reflect end-point achievement.

The standard of achievement each learner attains on each criterion is recorded as a rating of ‘C’ (satisfactory standard) according to the outcomes specified in the standards section of the course document.

A ‘t’ notation must be used where a learner demonstrates any achievement against a criterion less than the standard specified for the ‘C’ rating. The ‘t’ notation is not described in course standards.

A ‘z’ notation is to be used where a learner provides no evidence of achievement at all.
Internal assessment of all criteria will be made by the provider. Assessment processes must gather evidence that clearly shows the match between individual learner performance, the standards of the course and the learner’s award. Providers may report the learner’s rating for each criterion to TASC.

NOTATION ON EVIDENCE OF LEARNING

Some of the criteria standard elements use the terms reacting, responding and recognising.

In the context of this preliminary level course reacting/responding might be defined as:

• an intentional repeatable and observable behaviour that can be interpreted simply, objectively and conclusively.

In the context of this preliminary level course recognising might be defined as:

• intentionally signalling identification of (someone or something) through gesture, facial expression or sound from having encountered them before.

Each provider is responsible for ensuring the integrity and validity of their assessment results against the requirements of the course, including standards, and for maintaining records and documentation that will demonstrate the integrity, accuracy and validity of the award decisions they make each year.

Under Section 55ZN of the Office of Tasmanian Assessment, Standards and Certification Act 2003, TASC may give a provider written direction regarding the provision and assessment of an accredited other education course.

The assessment of Preliminary Mathematics Stage 3 will be based on whether the learner can:

• 1. Identify whole numbers and apply fundamental number skills to everyday situations
• 2. Use units of measurement
• 3. Use spatial awareness to describe position, location and shape
• 4. Collect, represent, and interpret statistical information

 

Criterion 1: Identify whole numbers and apply fundamental number skills to everyday situations

The learner:

Rating C
recognises and matches number names, numerals and quantities, up to 10 (including zero)
uses number names in sequence to count forwards and backwards in everyday situations from one to ten
subitises regular arrangements of objects and arrays up to 10 (e.g. dice, dominoes and card configurations)
adds two whole numbers up to a total of 10 or less
identifies a group of objects as being one (1) ‘more’ or ‘less’ than another group
compares and orders two collections according to their quantity (up to ten)
demonstrates in practical situations, ‘adding one more to’ and ‘taking one away from’
shares materials in practical situations using one-to-one correspondence
uses direct comparison to sort coins and notes into groups
adds two whole dollar monetary amounts up to a total of $10 or less
recognises ordinal numbers from 1st to 10th in written and spoken form.

Criterion 2: Use units of measurement

The learner:

Rating C
copies the use of simple measuring equipment for measuring temperature and time
selects appropriate simple measuring equipment to measure length, weight and capacity
compares two (or more) objects based on measurement attributes of length
measures length/width/height of objects or distances in everyday contexts using informal units (such as hand spans, toy car lengths)
responds to contexts involving ‘heavier/lighter’ than, ‘hotter/colder’ than and holds ‘more/less’ than
sorts objects according to a measurable attribute (e.g. weight) based on known physical characteristics
recognises the names of the days of the week
uses time markers to recognise sequence of events (e.g. before, after, next, then)
orders events based on time duration from least to most time taken
reacts to environmental time cues (e.g. sunset, high tide).

Criterion 3: Use spatial awareness to describe position, location and shape

The learner:

Rating C
sorts and organises common shapes into categories by name (e.g. triangles, rectangles, circles)
uses a range of terms to describe shapes (e.g. number of sides, same length)
gives and follows directions to move from one location to another and/or locate a familiar object not in sight
uses common location terms to describe situations involving movement or to locate an object/person (e.g. in the bag room, around the goal posts).

Criterion 4: Collect, represent, and interpret statistical information

The learner:

Rating C
collects information to answer a question (e.g. ‘what colours are the counters in the box?’)
identifies the choices/responses of a data display
follows simple instructions to sort objects / information into a simple data display.
matches objects relevant to a given context in a data display
uses key data in simple data displays to make decisions in everyday situations.

Preliminary Mathematics to Level 1: Stage 1, 2, 3 or 4 (with the award of):

SATISFACTORY ACHIEVEMENT

PRELIMINARY ACHIEVEMENT.

The final award for Preliminary Mathematics Stage 3 will be determined by the provider from four (4) ratings.

The minimum requirements for an award in each Stage of this course are as follows:

Satisfactory Achievement (SA)
4 ‘C’ ratings

Preliminary Achievement (PA)
1 ‘C’ rating

The Department of Education’s Curriculum Services will develop and regularly revise the curriculum. This evaluation will be informed by the experience of the course’s implementation, delivery and assessment. In addition, stakeholders may request Curriculum Services to review a particular aspect of an accredited course.

Requests for amendments to an accredited course will be forward by Curriculum Services to the Office of TASC for formal consideration. Such requests for amendment will be considered in terms of the likely improvements to the outcomes for learners, possible consequences for delivery and assessment of the course, and alignment with Australian Curriculum materials.

A course is formally analysed prior to the expiry of its accreditation as part of the process to develop specifications to guide the development of any replacement course.

The Department of Education acknowledges the significant leadership of Joshua Moore in the development of this course.

The accreditation period for this course is from 1 January 2019 until 31 December 2025. 

During the accreditation period required amendments can be considered via established processes. 

Should outcomes of the Years 9-12 Review process find this course unsuitable for inclusion in the Tasmanian curriculum, its accreditation may be cancelled. Any such cancellation would not occur during an academic year.

Version 1 - This course was accredited on 14 December 2018 for use from 1 January 2019 until 31 December 2023.

Version 2 - 28 April 2020. Corrections: addition of element 11 to Criterion 1; and elements 4 & 5 to Criterion 4.

Accreditation renewed on 22 June 2023 for the period 1 January 2024 until 31 December 2025 without amendment.

GLOSSARY OF VERBS

add	to find the sum of
arrange	place in proper, desired or convenient order
associate	combine or connect relating ideas
choose	decide or select the most suitable from a number of different options
classify	sort items into distinct groups with common elements
collect	to gather together
communicate	to impart knowledge of something through words, gestures or other non-verbal means
compare	to examine (two or more objects) in order to note similarities
copy	to follow as a pattern or model; imitate
count	to list or name the numerals up to (in order)
demonstrate	to impart knowledge of something through actions
describe	in the context of this course, 'describing' might be done via methods such as an oral description a written list of characteristics a series of actions a series of images
follow	act in accordance to instructions
give	impart or communicate instructions
identify	recognise the name or indicate who or what someone or something is
indicate	show through verbal or non-verbal means
interpret	to bring out the meaning of
label	identify by placing a name, word or picture used to describe an object or attribute
make	put together
match	to cause to correspond (e.g. name with quantity)
model	explain using materials
name	identify as proper/common noun
organise	group according to system
react	a deliberate, repeatable and observable behaviour that can be interpreted simply, objectively and conclusively
recognise	be aware of or acknowledge
respond	provide an answer; reply
select	choose in preference to another or others
sequence	arrange in order
share	divide and distribute
sort	arrange by feature
subitise	to perceive at a glance the number of items presented
use	to employ for some purpose

 

GLOSSARY – KEY TERMS

area	a measure of how many units are required to cover a surface. The units are usually standard units, such as square centimetres or square metres
axis	in the given context, axis is a function of a graph as a fixed reference in conjunction with another fixed reference for determining the position of a point or series of points
capacity	in the given context,capacity is a term that describes how much a container will hold
data	a general term for information (observations and/or measurements) collected during any type of systematic investigation
direction	in the given context, direction refers to the line along which anything lies, faces, moves etc.
fraction	a number expressed in the form, representing a part divisible by a whole
frequency table	a data display to list possible outcomes and show the number of times each outcome occurred in a given context e.g.
location	a particular place or position
mass	the measure of how much matter is in a person, object, or substance
model	a standard or example for imitation or comparison
one-to-one correspondence	refers to the act of simultaneously counting in sequence by saying number names and selecting/identifying the correct number of objects e.g. a child not using one-to-one correspondence when counting objects will not co-ordinate saying the number names with gesturing to the objects and may say more than one number per object or miss objects
order	in the given context order refers to the arrangement of numbers in relation to each other according to a particular sequence, pattern, or method
ordinal	of or relating to order, rank, or position in a series
partition	a division into or distribution in portions or shares
pattern	a regular and intelligible form or sequence discernible by following a rule or rules
picture graph	a statistical graph for organising and displaying categorical data e.g.
place value	the value of a digit as determined by its position in a number, relative to the ones (or units) place. For integers, the ones place is occupied by the rightmost digit in the number
position	a particular way in which someone or something is placed or arranged
scale	in the given context, scale refers to a series of marks laid down at determinate distances, as along a line, for purposes of measurement
sequence	the following of one thing after another
subitise	to perceive at a glance the number of items presented
survey	in the given context,a survey refers to the sampling or partial collection, of facts, figures, or opinions taken and used to approximate or indicate what a complete collection and analysis might reveal
three-dimensional	an object is three-dimensional when it possesses the dimensions of height, width and depth. Two dimensional objects only have two dimensions: length and width. A solid is any geometrical object with three-dimensions
two-dimensional	a shape is two-dimensional when it only possesses the dimensions of length and width
volume	a measure of the space enclosed by a solid. For a rectangular prism, Volume=Length×Width×Height

Learning Outcome	Criterion	Content	Criterion and elements
identify whole numbers and apply fundamental number skills to everyday mathematical situations	Criterion 1	Stage 1: Number and algebra Stage 2: Number and algebra Stage 3: Number and algebra Stage 4: Number and algebra	Stage 1 C1 All elements Stage 2 C1 All elements Stage 3 C1 All elements Stage 4 C1 All elements
use units of measurement	Criterion 2	Stage 1: Measurement and geometry Stage 2: Measurement and geometry Stage 3: Measurement and geometry Stage 4: Measurement and geometry	Stage 1 C2 All elements Stage 2 C2 All elements Stage 3 C2 All elements Stage 4 C2 All elements
use spatial awareness to describe position, location and shape	Criterion 3	Stage 1: Measurement and geometry Stage 2: Measurement and geometry Stage 3: Measurement and geometry Stage 4: Measurement and geometry	Stage 1 C3 All elements Stage 2 C3 All elements Stage 3 C3 All elements Stage 4 C3 All elements
collect, represent and interpret statistical information	Criterion 4	Stage 1: *No content Stage 2: *No content Stage 3: Statistics and probability Stage 4: Statistics and probability	Stage 1 *No assessment Stage 2 *No assessment Stage 3 C5 All elements Stage 4 C5 All elements