What is a mathematics exemplar?
A mathematics exemplar is an annotated sample of student work produced in response to a set task. Each exemplar illustrates student work based on a particular topic and strand of Mathematics in the New Zealand Curriculum. They are based on authentic examples of students' work, chosen from thousands of samples collected from schools throughout New Zealand.
Each curriculum strand, but not each achievement objective for that strand, is exemplified. The mathematics exemplars relate to curriculum levels 1 to 5.
The seven sets of mathematics exemplars cover:
The level 1 mathematics exemplars make a link to early childhood learning through Te Whāriki: He Whāriki Mātauranga mō ngā Mokopuna o Aotearoa/Early Childhood Curriculum. Similarly, the level 5 exemplars suggest a link to a related achievement standard in the National Certificate of Educational Achievement (NCEA).
Two things underpin the mathematics exemplars: progressions and tasks.
Progressions
With each exemplar set there is an introduction to the progression that summarises the whole set and identifies the task or tasks exemplified in the student work.
Each exemplar set illustrates a progression in mathematical learning. For example, in measurement, students progress through direct comparison, indirect comparison, use of non-standard units, use of standard units, to flexibility in applying the methods illustrated at each previous step.
The steps in each mathematics progression are the major ones that students move through in learning about that topic. Not every student will move through each of these steps in the given order, although most will do so. In practice, students may be able to respond at more than one level in a progression. It is important for the teacher to determine the highest level of the progression at which a student can perform. Each exemplar includes teacher-student conversations that suggest scaffolds to elicit a student's most sophisticated level of thinking and their currently highest step in that progression.
The seven progressions represented in the exemplar sets are set out in the mathematics matrix.
The choice of progressions for the mathematics exemplars was guided by the need to:
- ground the progressions on research based on samples of work from schools throughout New Zealand
- illustrate the main content areas of Mathematics in the New Zealand Curriculum
- provide material that will help teachers to assess achievement in mathematics and provide feedback to their students
- support areas of mathematics in which students are known to experience some difficulty.
The exemplar sets are linked to the following content areas of mathematics:
- identifying and ordering numbers – fractions set
- operating with numbers – number strategy set
- patterns in number – exploring patterns set
- measurement – measuring set
- identifying shapes and their properties – tessellations set
- operating with shapes – tessellations set
- statistical investigations – data display set
- probability – probability set.
These content areas can be related to:
While each set of exemplars is linked to one of these content areas, it does not provide a complete coverage of the mathematical understandings relevant to that area.
Tasks
The key to the mathematics exemplars is the task or tasks on which each exemplar set is based. These tasks will help teachers to determine the highest level of the progression at which a student is able to perform, and to move the student onwards through the progression. This intention reflects the philosophy and diagnostic interviews of the Numeracy Professional Development Projects. In the various exemplar sets, the task may be the same for several steps of that progression, or it may be expressed in increasingly difficult versions at each step.
It is hoped that teachers will be able to generalise the tasks to other mathematics topic areas and use them as models to generate their own exemplars.
The rich tasks require a relatively short time to administer. Teachers should be able to incorporate them easily into their regular mathematics programme. The tasks give students opportunities to perform at a number of levels.
Number
Fractions
In responding to the tasks in this exemplar set, students move from identifying and stating the fractions of regions, to comparing fractions, and then to understanding the link between fractions and decimals.
This topic was chosen because the understanding and use of fractions is an identified area of weakness in the achievement of New Zealand students. It is particularly important for teachers to assess their students' understanding of fractions and to help them move to the next level.
Fractions are important in relation to at least two further basic areas of the curriculum: probability and algebra. Knowledge of fractions is required before students can begin to work on numerical probability problems. As students' algebraic skills develop, they need to use "algebra" fractions in certain manipulations in order to solve equations.
Number strategies
The tasks in this set display the mental strategies that students use to solve number problems. In responding to the tasks in this exemplar set, students move from a single-digit addition problem that can be solved by a counting-from-one strategy, to a ratio problem that they might solve using an advanced proportional part-whole strategy.
This set of exemplars was chosen to build on work in the Numeracy Professional Development Projects and link it to the New Zealand Curriculum Exemplars Project. The drawing together of these major projects in the number strategy exemplars may help teachers to better understand the relationship between two important government numeracy initiatives.
Measurement
Measuring
A simple question (or task) was chosen for this progression: "Will the table fit through the door?"
Although the specific focus in this task is length, the task and the principle it embodies can be transferred to other aspects of measurement. The same progression: direct comparison, indirect comparison, non-standard units, standard units, flexibility, is also applicable in two and three dimensions to measuring areas and volumes. It can also be applied to measuring weights and, to a lesser extent, to measuring time and temperature.
Geometry
Tessellations
The geometry task is based on tessellating shapes. Students are asked which of a variety of shapes tessellate and why. At level 1, students recognise basic shapes and how they do or do not fit together. At level 5, students are able to find the interior angles of polygons to determine whether particular angles are divisors of 360 degrees.
The topic of tessellations was selected because it allows teachers to use a simple task and situation to determine a wide range of geometrical knowledge.
Algebra
Exploring patterns
The task in this progression is based on a relatively sophisticated pattern involving red and blue counters. Students move from finding the next element in the pattern to using an algebraic equation to solve a variety of problems related to that pattern.
The task in this progression is sufficiently complex that all five curriculum levels can be illustrated; indeed, level 5, in this exemplar set only, is divided into two sublevels: 5i and 5ii. This exemplar set also shows students progressing from manipulation of whole numbers to the use of algebraic symbols and equations. This is an important transition for teachers to understand. It shows them that a student's proficiency in number and number strategies is a precursor to algebraic thinking.
Statistics
Data display
The increasingly difficult versions of the task in this exemplar set generate a range of methods for displaying data, from pictograms to stem-and-leaf graphs and box-and-whisker plots.
This topic was chosen because of its importance to modern society. Data is presented in every area of the media in a variety of formats and displays. In this century, everyone needs to be able to scan and understand data displays in order to make good sense of the many arguments and debates in the media or in other reading or studies.
Generation or interpretation of data displays is an important aspect of study in other essential learning areas.
Probability
The core task in the probability exemplar set involves the chance of drawing a red or a blue ball from buckets, each of which contains a different combination of these colours. The task is expressed in increasingly complex versions to elicit a student's position in the progression. At level 1/2, the student understands and can express simple ideas of chance, using terms such as "certain", "possible", or "impossible". The student at level 5 is able to understand more complicated ideas of probability and express them by using, for example, probability trees.
International research has demonstrated that the concept of probability is difficult for students to learn and for teachers to teach. The average person has a poor understanding of all aspects of probability, yet probability is important in the everyday world. Probability applications include estimating and making choices about whether or not to do something in given circumstances. Applications range from deciding whether to buy a raffle ticket, to an insurance actuary's job of determining the likelihood of (mainly catastrophic) events and setting premiums accordingly.
The topic of probability was chosen in the hope that a better understanding in the classroom will encourage the use of probability applications in daily life and raise public understanding of this important concept.
Using the exemplars
Teachers are likely to use the exemplars:
- with whole classes, groups, or individuals to illustrate the big ideas of mathematics and to inform assessment practices
- as models to draw students' attention to important features of their work
- as a scaffold to support student learning in areas of difficulty
- to help inform parents, board members, and principals about what students are achieving in mathematics, how they are progressing, and what their next learning steps will be.
The New Zealand Curriculum Exemplars Online
Access the mathematics exemplars at www.tki.org.nz/r/assessment/exemplars/maths/
Links are provided to other essential materials, including relevant curriculum documents and the NCEA achievement standards on the website for the National Certificate of Educational Achievement.
The mathematics exemplar materials are available in both PDF and HTML format.
Print teachers' notes (PDF, 47kb)
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