National Numeracy Learning Progression - Australian Curriculum
National Numeracy LearningProgressionCOPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence. See(creativecommons.org/licenses/by-nc/4.0/). You are free to share (copy and redistribute the material in any medium or format) and adapt (remix, transform, andbuild upon) these materials for non-commercial purposes only, provided you attribute ACARA. See attribution notice under our Copyright and terms of s-of-use/). For the avoidance of doubt, this means that you must not use these materials for commercial purposes.
ContentsWhat is numeracy? . 3What is the focus of the numeracy progression? . 3How is the numeracy progression structured? . 3How can the numeracy progression be used? . 6Number sense and algebra . 7Quantifying numbers . 7Additive strategies . 13Multiplicative strategies . 15Operating with decimals . 18Operating with percentages . 20Understanding money . 22Number patterns and algebraic thinking . 24Comparing units (ratios, rates and proportion) . 27Interpreting fractions . 29Measurement and geometry . 31Understanding units of measurement . 31Understanding geometric properties . 35Positioning and locating . 37Measuring time . 38Statistics and probability . 40Understanding chance. 40Interpreting and representing data . 42COPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence. See(creativecommons.org/licenses/by-nc/4.0/). You are free to share (copy and redistribute the material in any medium or format) and adapt (remix, transform, andbuild upon) these materials for non-commercial purposes only, provided you attribute ACARA. See attribution notice under our Copyright and terms of s-of-use/). For the avoidance of doubt, this means that you must not use these materials for commercial purposes.
What is numeracy?Numeracy is fundamental to a student’s ability to learn at school and to engage productively insociety.In the Australian Curriculum, students become numerate as they develop the knowledge andskills to use mathematics confidently across learning areas at school and in their lives morebroadly. The Australian Curriculum states:Numeracy encompasses the knowledge, skills, behaviours and dispositions that students need to usemathematics in a wide range of situations. It involves students recognising and understanding the role ofmathematics in the world and having the dispositions and capacities to use mathematical knowledge andskills purposefully (ACARA 2017).What is the focus of the numeracy progression?Numeracy development influences student success in many areas of learning at school. Theprogression can be used to support students to successfully engage with the numeracydemands of the Foundation to Year 10 Australian Curriculum.The National Numeracy Learning Progression outlines a sequence of observable indicators ofincreasingly sophisticated understanding of and skills in key numeracy concepts. By providing acomprehensive view of numeracy learning and how it develops over time, the progression givesteachers a conceptual tool that can assist them to develop targeted teaching and learningprograms for students who are working above or below year-level expectations.The progression does not advise on how to teach, plan, program, assess or report in schools. Itrecognises the importance of, but does not describe, the sequence for specific learning areacontent related to numeracy development such as graphing and constructing timelines.The Australian Core Skills Framework has been used to guide decisions on the scope of theprogressions. The progression is designed to assist students in reaching a level of proficiency inliteracy to at least Level 3 of the Core Skills Framework.How is the numeracy progression structured?Elements and sub-elementsThe National Numeracy Learning Progression has three elements that reflect aspects ofnumeracy development necessary for successful learners of the F–10 Australian Curriculum andin everyday life. The three elements are: Number sense and algebraMeasurement and geometryStatistics and probability.Each of the elements includes sub-elements that present developmental sequences forimportant aspects of numeracy capability. There are nine sub-elements in Number sense andalgebra, four in Measurement and geometry and two in Statistics and probability.The diagram (Figure 1) represents the elements and sub-elements in relation to the numeracydevelopment of the student.3COPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence.See (creativecommons.org/licenses/by-nc/4.0/). For more information, please see the first page of these materials and our Copyright and terms of s-of-use/).
Figure 1. Elements and sub-elements of the National Numeracy Learning ProgressionLevels and indicatorsWithin each sub-element indicators are grouped together to form developmental levels. Eachindicator describes what a student says, does or produces and begins with the implicit stem‘A student ’ as the subject of the sentence.There are as many levels within each sub-element as can be supported by evidence. Thelisting of indicators within a level is non-hierarchical as the levels are collections ofindicators. Each level within a sub-element has one or more indicators and is moresophisticated or complex than the preceding level.In many of the sub-elements, subheadings have been included to assist teachers bygrouping indicators into particular categories of skills that develop over a number of levels.The amount of time it takes students to progress through each level is not specified sincestudents progress in numeracy development at different rates.The levels do not describe equal intervals of time in students’ learning. They are designedto indicate the order in which students acquire the knowledge and skills necessary to be4COPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence.See (creativecommons.org/licenses/by-nc/4.0/). For more information, please see the first page of these materials and our Copyright and terms of s-of-use/).
numerate. As learning is very rapid in the early years of school, the initial levels tend to bemore detailed than the later levels.Moreover, the amount of detail in any level or sub-element is not an indication ofimportance. A single indicator at a higher level in the progression may rely on a substantialnumber of indicators being evident in earlier levels. The diagram (Figure 2) shows thevarious components included in the progression.Figure 2. Annotated example of a numeracy sub-elementHow is the numeracy progression related to the Australian Curriculum?Numeracy skills are explicit teaching in the Australian Curriculum: Mathematics. Studentsneed opportunities to recognise that mathematics is constantly used outside themathematics classroom and that numerate people apply general mathematical skills in awide range of familiar and unfamiliar situations.Using mathematical skills across the curriculum enriches the study of other learning areasand helps to develop a broader and deeper understanding of numeracy. It is essential thatthe mathematical ideas with which students interact are relevant to their lives.Australian Curriculum: MathematicsThe Australian Curriculum: Mathematics provides students with essential mathematical skills andknowledge in number and algebra, measurement and geometry, and statistics and probability Mathematics is composed of multiple but interrelated and interdependent concepts and systemswhich students apply beyond the mathematics classroom (Australian Curriculum: Mathematics,Rationale 2017)The Australian Curriculum: Mathematics sets teaching expectations for mathematicslearning at each year level, providing carefully paced, in-depth study of critical mathematicalskills and concepts. The curriculum focuses on developing the mathematical proficiencies of5COPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence.See (creativecommons.org/licenses/by-nc/4.0/). For more information, please see the first page of these materials and our Copyright and terms of s-of-use/).
understanding, fluency, reasoning and problem solving. These proficiencies are reflected inthe National Numeracy Learning Progression rather than specifically identified.The National Numeracy Learning Progression helps teachers to develop fine-grainunderstandings of student numeracy development in the Australian Curriculum:Mathematics, especially in the early years. It is particularly useful in guiding teachers tosupport students whose numeracy development is above or below the age-equivalentcurriculum expectations of the Australian Curriculum: Mathematics. The progression hasnot been designed as a checklist and does not replace the Australian Curriculum:Mathematics.Each sub-element has been mapped to the year-level expectations set by the AustralianCurriculum: Mathematics.Other Australian Curriculum learning areasThis National Numeracy Learning Progression is designed to assist schools and teachersin all learning areas to support their students to successfully engage with the numeracydemands of the F–10 Australian Curriculum.Advice is included on identifying the numeracy demands of each subject in the AustralianCurriculum. This advice will assist teachers to identify opportunities to support students’numeracy development and to provide meaningful contexts for the application ofnumeracy skills.How can the numeracy progression be used?The National Numeracy Learning Progression can be used at a whole school, team orindividual teacher level. However, the progression provides maximum student learningbenefits when used as part of a whole-school strategy that involves professional learningand collaboration between teachers. Further advice on how to maximise the benefits of theprogression is available on the progressions home page.The numeracy progression can be used to identify the numeracy performance of individualstudents within and across the 15 sub-elements. In any class there may be a wide rangeof student abilities. Individual students may not neatly fit within a particular level of theprogressions and may straddle two or more levels within a progression. While theprogression provides a logical sequence, not all students will progress through every levelin a uniform manner.When making decisions about a student’s numeracy development, teachers selectrelevant indicators. It is important to remember indicators at a level are not a prescriptivelist and the progression is not designed to be used as a checklist. Teacher judgementsabout student numeracy capability should be based on a range of learning experiences.Number talks, written or oral explanations, or tasks from any learning area can providesuitable evidence of a student’s numeracy capability.Teachers can use the progressions to support the development of targeted teaching andlearning programs and to set clearer learning goals for individual students. For example,6COPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence.See (creativecommons.org/licenses/by-nc/4.0/). For more information, please see the first page of these materials and our Copyright and terms of s-of-use/).
teaching decisions can be based on judgements about student capability that relate to asingle indicator rather than all indicators at a level.Number sense and algebraQuantifying numbersAlthough number is an abstract concept which can be represented by a word, a symbol(numeral) or an image, it is central to quantitative thinking.This sub-element describes how a student becomes increasingly able to count, recognise,read and interpret numbers expressed in different ways. It outlines key understandingsneeded to process, communicate and interpret numerical information in a variety ofcontexts.Within this sub-element, place value is taken to mean more than being able to read, writeand state the positional value of a digit. Place value relies on understanding therelationship between digits in a numeral, which then enables the numeral to be renamed inmultiple ways. In addition to the base-ten positional value property, the place value systemhas both additive and multiplicative properties. That is, the quantity represented by anumeral is the sum of the values represented by its individual digits (326 300 20 6)and the value of a digit is determined by multiplying its face value by the value assigned toits position in the numeral (326 3 x 100 2 x 10 6 x 1).The Quantifying numbers sub-element underpins learning of number sense, measuringand using data.Some students will communicate using augmentative and alternative communicationstrategies to demonstrate their numeracy skills. This may include digital technologies, signlanguage, braille, real objects, photographs and pictographs.LevelIndicatorsEach sub-element level has been identified by upper-case initials and in some cases lower-case letters of thesub-element name followed by ascending numbers. The abbreviation for this sub-element is QuN. The listingof indicators within each level is non-hierarchical. Subheadings have been included to group related indicators.Where appropriate, examples have been provided in brackets following an indicator.Producing number names QuN1produces number words that relate to students’ lives, which couldinvolve the use of augmentative and alternative communication (AAC)Counting items responds to a request for a different amount by increasing or decreasinga quantity7COPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence.See (creativecommons.org/licenses/by-nc/4.0/). For more information, please see the first page of these materials and our Copyright and terms of s-of-use/).
Quantifying numbers recognises the effects of adding to and taking away from a collection ofobjectsNumber recognition and identification recognises small quantities ( 4) as being the same or different withoutcounting (subitises)compares two quantities and states which group has more and whichlessmatches one numeral with another (matches to a sample)recognises some numerals, such as those associated with age or homeaddressProducing number names produces a rote count to at least 12 produces a rote count down from 10Counting itemsQuN2 counts a small number of items (typically less than 4)Numeral recognition and identification indicates the correct numeral from a range of different symbols for mostnumerals up to 10 (‘which is 3?’)produces the matching number word for most numerals up to 10Producing number names produces the number word just after a given number word in the range1–10 (but drops back to 1 when doing so)produces the number word just before a given number word in the range1–10 (but drops back to 1 when doing so)Counting itemsQuN3 recognises that the last number word said in a count answers ‘Howmany?’matches the count (up to 10) to objects, using the one-to-one principleNumeral recognition and identification recognises and identifies all numerals in the range 1–10selects the largest numeral from an unordered group of 3 or more, in therange 1–10 Reference to producing number names to at least 120 rather than 100 is because of the higher proportion ofstudents in the early years who encounter a hurdle at 109 compared to 100.8COPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence.See (creativecommons.org/licenses/by-nc/4.0/). For more information, please see the first page of these materials and our Copyright and terms of s-of-use/).
Quantifying numbersProducing number names produces the number word just after a given number word in the range1–10 (without dropping back to count from 1)produces the number word just before a given number word in the range1–10 (without dropping back)Counting itemsQuN4 matches number words within the current known counting range toquantities of itemscorrectly indicates the larger or smaller of two numerals in the rangefrom 1 to 10Numeral recognition and identification recognises and identifies all numerals in the range 1–10 as well as 20,30, 40, 50, 60, 70, 80, 90 and 100orders numerals to at least 10Producing number names QuN5counts to at least 20continues a count from a number other than 1counts forwards by tens to 100Counting items counts groups of up to 20 itemsNumeral recognition and identification points to the correct numeral in response to a verbal request, fornumerals up to 20 as well as 30, 40, 50, 60, 70, 80, 90 and 100Producing number names QuN6 counts to at least 30produces the number word just after a given number in the range 1–30(without dropping back)produces the number word just before a given number word in the range1–30 (without dropping back)counts forwards and backwards by tens to and from 100Counting items matches known numerals (to 20) to quantities9COPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence.See (creativecommons.org/licenses/by-nc/4.0/). For more information, please see the first page of these materials and our Copyright and terms of s-of-use/).
Quantifying numbersNumeral recognition and identification identifies all numerals up to 30 as well as 40, 50, 60, 70, 80, 90 and 100(is shown the numeral 17 and produces its name)orders numbers to at least 20 (determines the largest number in a groupof numbers selected from 1 to 20)Producing number names to at least 120 counts forwards and backwards to and from 120 and beyondcontinues counting from any number up to 120 and beyondcounts forwards and backwards by fivesGrouping and counting items by tensQuN7 counts items in groups of twos, fives and tensrecognises that a count of one ten is the same as ten counts of oneNumeral recognition and identification identifies numerals from 0 to at least 100 (is shown the numeral 45 andproduces its name)recognises a numeral from a given range up to 100 (is shown thenumerals 70, 38, 56 and 26 and when asked which is 38, indicates thenumeral)Producing number names to at least 1000 counts forwards and backwards by 100s to 1000 (100, 200 1000)counts forwards and backwards by tens off the decade to 100 (2, 12, 22, )Numeral recognition and identification of place valueQuN8 recognises and describes teen numbers as 1 ten and some more (16 is1 ten and 6 more)represents and renames two-digit numbers as separate tens and ones(68 is 6 tens and 8 ones, 68 ones, or 60 8)applies an understanding of zero in place value notation when readingnumerals that include internal zeros (correctly recognises 101 as onehundred and one, not 1001)Producing number names of any sizeQuN9 counts forwards and backwards from any numberproduces and reads numbers to at least 1000 Reference to producing number names to at least 120 rather than 100 is because of the higher proportion ofstudents in the early years who encounter a hurdle at 109 compared to 100.10COPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence.See (creativecommons.org/licenses/by-nc/4.0/). For more information, please see the first page of these materials and our Copyright and terms of s-of-use/).
Quantifying numbersNumeral recognition and identification of place value recognises and identifies numerals from a given range up to 1000 (isshown the numerals 170, 318, 576 and 276 and when asked which is276, points to the 276)Understanding place value represents and flexibly renames three-digit numbers as counts ofhundreds, tens and ones (247 is 2 hundreds, 4 tens and 7 ones, or 2hundreds and 47 ones, or 24 tens and 7 ones)Understanding decimal place value recognises that the place value system can be extended to tenths andhundredthsuses an understanding of the magnitude of decimals to compare them totwo decimal places (0.20 is smaller than 0.4)orders decimals to one decimal place by placing them on an intervalbetween 0 and 1Numeral recognition and identification of place value identifies numerals in the range 0–10 000 (is shown the numeral 2001and responds two thousand and one)recognises a numeral from a given range of numerals up to 10 000(when presented with the numerals 1701, 9318, 2050 and 2500 andwhen asked which is 2050, indicates the correct numeral)Understanding place value QuN10 reads and writes numbers beyond 1000 applying knowledge of the placevalue periods of ones and thousandspartitions numbers by their place value into thousands, hundreds, tensand onesUnderstanding decimal place value locates and orders decimals between 0 and 1 up to two decimal placesrecognises that the place value system can be extended to thousandthscompares the size of decimals (including ragged decimals such as 0.5,0.25, 0.125)reads, compares and renames decimal numbers (0.4 is greater than0.355 because 0.4 has 4 tenths and 0.355 only has 3 tenths)Understanding place valueQuN11 reads and writes numbers applying knowledge of the place valueperiods of ones, thousands, millions (how numbers are written with thedigits organised in groups of three – 10 000 is read as ten thousand,where thousand is the place value period)11COPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence.See (creativecommons.org/licenses/by-nc/4.0/). For more information, please see the first page of these materials and our Copyright and terms of s-of-use/).
Quantifying numbers partitions numbers by their place value into tens of thousands,thousands, hundreds, tens and ones and beyondrecognises the relationship between adjacent positions in place value(200 is 10 times as large as 20, which is 10 times as large as 2)estimates whole numbers to the nearest hundred thousand, tenthousand, etc. (crowd numbers at a football match)Understanding decimal place value compares and orders decimals beyond 1 including ragged decimals(those expressed with unequal numbers of places)recognises the relationship between adjacent positions in place value fordecimals (0.20 is 10 times larger than 0.02)Understanding place value (directed numbers) orders negative numbers (recognises that 10 C is colder than 2.5 C)Representing place valueQuN12 recognises, reads and interprets very large and very small numbersexpresses numbers as powers of 10 in scientific notation anddetermines the order of magnitude of quantities (a nanometre has anorder of magnitude of 9)relates place value parts to indices (1000 is 100 times larger than 10,and that is why 101 x 102 103 and why 103 divided by 101 is equal to102)12COPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence.See (creativecommons.org/licenses/by-nc/4.0/). For more information, please see the first page of these materials and our Copyright and terms of s-of-use/).
Additive strategiesThis sub-element describes how a student becomes increasingly able to choose and useadditive computational strategies for different purposes. The transition from counting byone to more flexible methods of dealing with quantity, where numbers are treated as thesums of their parts, is a critical hurdle to be addressed in students becoming fluent usersof number. Rather than only focusing on the speed of producing correct answers, anemphasis on attending to the relation of given numbers to sums and differences is neededfor flexibility. This supports the development of additive strategies such as adding thesame to both numbers to reach an easier calculation (47 – 38 49 – 40).The capacity to make reasonable adjustments to numbers is essential in estimating.Estimating is not a basic skill as it requires students to be able to conceptualise andmentally manipulate numbers. The estimation process involves selecting numbers tosimplify mental manipulation.Additive strategies apply equally to subtraction, as can be seen in ‘Giving change’ in theUnderstanding money sub-element.Some students will communicate using augmentative and alternative communicationstrategies to demonstrate their numeracy skills. This may include digital technologies, signlanguage, braille, real objects, photographs and pictographs.LevelIndicatorsEach sub-element level has been identified by upper-case initials and in some cases lower-case letters of thesub-element name followed by ascending numbers. The abbreviation for this sub-element is AdS. The listing ofindicators within each level is non-hierarchical. Subheadings have been included to group related indicators.Where appropriate, examples have been provided in brackets following an indicator.Emergent strategiesAdS1 combines two groups of objects and attempts to find the totalcompares two quantities of up to 10 and states which group has morePerceptual strategiesAdS2 counts items that can be perceived by ones to find the total of twogroups with one-to-one matching of number words and objectsbuilds and subtracts numbers by using objects or fingersmakes combinations to form numbers up to 10Figurative (imagined units)AdS3 solves additive tasks involving two concealed collections of items byvisualising, counting from one to determine the total13COPYRIGHTThe National Literacy Learning Progressions are licensed under a creative commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) licence.See (creativecommons.org/licenses/by-nc/4.0/). For more information, please see the first page of these materials and our Copyright and terms of s-of-use/).
Additive strategiesCounting on (by ones) AdS4 treats a number word as a completed count when solving problems (‘Ihave 7 apples. I want 10. How many more do I need?’ Treats the 7 as acompleted count)uses a strategy of count-up-from to calculate addition (to find 6 3,responds 6, 7, 8, 9. It's 9)uses a strategy of count-up-to to solve missing addends tasks (to solve6 ? 9, responds 6 . 7, 8, 9. It's 3)Counting back (by ones) AdS5 uses count-down-from for subtraction tasks (9 – 3 ?, 9 . 8, 7, 6. Itequals 6)uses count-down-to to calculate (9 take away something equals 6,responds 9 . 8, 7, 6 . It's 3)finds the difference between two numbers less than 20counts back to find the difference between two quantities where thedifference is no greater than 4Flexible strategies with combinations to 10 AdS6 uses a range of non–count-by-one strategies when adding or subtractingtwo or more numbers (bridging to 10, near doubles)uses part-whole construction of number to partition a whole number intoparts (partitions 7 into 5 and 2, 6 and 1, 4 and 3)applies inverse r
Mathematics is composed of multiple but interrelated and interdependent concepts and systems which students apply beyond the mathematics classroom (Australian Curriculum: Mathematics, Rationale 2017) The Australian Curriculum: Mathematics sets teaching expectations for mathematics learning at each year level, providing carefully paced, in .
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Learning Pathways in Numeracy: Addressing Early Numeracy Skills Except where otherwise noted, content in this document is licensed by the Office of Superintendent of Public Instruction under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
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