Preliminary Mathematics Stage 4

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 4 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.

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 4 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 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 4 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 4: (50, 100 AND 150 HOURS)

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

In Stage 4, learners engage, explore and respond to experiences through communication with known adults, teachers and peers.
Within number and algebra, learners will identify place value for ones and tens and use this knowledge to perform addition of whole numbers totalling 100 or less. They will share objects in practical situations (without remainder) and use ordinal terminology to describe position (1st, 2nd, 3rd etc.). Learners will engage with the full suite of Australian coins and notes and sum multiple costs to a value of $100 or less in practical situations. 

Learners will:

• play games involving dice, playing cards and dominoes that require recognition of quantities and counting both forwards and backwards (e.g. snap, snakes and ladders)
• participate in songs, rhymes, games and stories that require counting forwards and backwards
• model situations of ‘taking away’ and describe the process in various ways including using the subtraction symbol
• explore all of the different ways that two whole number can be added to make 10 (with or without a calculator)
• compare and order collections using the appropriate language and number name
• explore stories and everyday situations which require sharing and the distribution of materials
• matching coins to 2-Dimensional images
• using money in ‘exchange of goods for cash’ simulations.

Within measurement and geometry, learners will sort and classify common regular shapes through identification. They will describe these shapes using a range of terms relating to specific attributes such as side length and the number of sides. They will explore the features of measurement instruments, and with support, interpret the readings obtained including recognition of the formal units. They will recognise milestone dates and sequence days of the week. They will identify digital time associated with routine daily events (e.g. school start time), categorise events by duration (short/long) and sequence routine tasks in order. 

Learners will:

• explore the relationship between shape and template involving 2-Dimensional shapes and 3-Dimensional objects (e.g. completing puzzles with linear and non-linear 2-Dimensional shapes, placing wooden objects into a box through corresponding holes cube through square, triangular prism through triangle)
• construct 3-Dimensional objects using templates and malleable materials
• experience reading measurements on linear scales (e.g. thermometer, oven dial, ruler, tape measure) and assist to use these measurements in more complex activities involving them
• conduct measurements to perform activities such as simple science experiments, design tasks, cooking, or planning a garden plot layout
• participate in activities to identify and sequence days of the week (e.g. songs, chants, stories)
• discuss milestone dates and events (e.g. birthdays, Christmas Day, lunch time, the weekend) using time markers including yesterday, earlier today, next week, and next month
• participate in games, dances and stories that involve sequences of movement
• participate in simple treasure hunts / orienteering exercises using sequenced movement or location steps (e.g. ‘next to the netball goal’, ‘under the oak tree’, ‘3 steps forward’).

Within statistics and probability, learners will assist in planning the data collection process of an investigation and carry out the collection individually or in groups. They will choose and justify the best type of display to represent the data according to the context. They can identify features of simple tables, graphs and charts and can create a simple graph using information obtained in a table or vice versa. They will interpret simple data displays obtained by others including newspaper reports, sport score sheets and media claims.

Learners will:

• collect data for relevant contexts (e.g. ‘How much plastic is generated from purchases at the school canteen each week?’)
• identify possible answers to a survey question (e.g. ‘How many times a week do you eat vegetables?’)
• match responses which are the same to assist in data collation
• identify which response is the largest / smallest
• predict what picture symbols might mean on data sources such as weather charts, maps, library charts
• make simple statements about data they have collected (e.g. dark chocolate is more popular than white chocolate)
• assign ordinal rankings (1st, 2nd, 3rd) to participants in a competition.

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 4 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

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 4 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 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 measurement and geometry learning outcome (#1); data fix to ensure Criterion 1 displays; element 10 added to Criterion 2; element 6 added to Criterion 3; and element 5 added to Criterion 4.

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

GLOSSARY OF VERBS

GLOSSARY – KEY TERMS