Mathematical Proficiencies

[Francisco Raya]

TheAustralian Curriculum: Mathematics aims to be relevant and applicable to the 21st century. The inclusion of the proficiencies of understanding, fluency, problem-solving and reasoning in the curriculum is to ensure that student learning and student independence are at the centre of the curriculum. The curriculum focuses on developing increasingly sophisticated and refined mathematical understanding, fluency, reasoning, and problem-solving skills. These proficiencies enable students to respond to familiar and unfamiliar situations by employing mathematical strategies to make informed decisions and solve problems efficiently.

Students develop skills in choosing appropriate procedures; carrying out procedures flexibly, accurately, efficiently and appropriately; and recalling factual knowledge and concepts readily. Students are fluent when they calculate answers efficiently, when they recognise robust ways of answering questions, when they choose appropriate methods and approximations, when they recall definitions and regularly use facts, and when they can manipulate expressions and equations to find solutions.

Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable.

Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. Students are reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning from one context to another, when they prove that something is true or false, and when they compare and contrast related ideas and explain their choices.

Adding It Up explores how students in pre-K through 8th grade learn mathematics and recommends how teaching, curricula, and teacher education should change to improve mathematics learning during these critical years.

The committee identifies five interdependent components of mathematical proficiency and describes how students develop this proficiency. With examples and illustrations, the book presents a portrait of mathematics learning:

The committee discusses what is known from research about teaching for mathematics proficiency, focusing on the interactions between teachers and students around educational materials and how teachers develop proficiency in teaching mathematics.

You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

Switch between the Original Pages, where you can read the report as it appeared in print, and Text Pages for the web version, where you can highlight and search the text.

The proficiencies of Understanding, Fluency, Problem Solving and Reasoning are fundamental to learning mathematics and working mathematically and are applied across all three strands Number and Algebra, Measurement and Geometry, and Statistics and Probability.

Fluency describes students developing skills in choosing appropriate procedures, carrying out procedures flexibly, accurately, efficiently and appropriately, and recalling factual knowledge and concepts readily. Students are fluent when they:

Problem-solving is the ability of students to make choices, interpret, formulate, model and investigate problem situations, select and use technological functions and communicate solutions effectively. Students pose and solve problems when they:

Reasoning refers to students developing an increasingly sophisticated capacity for logical, statistical and probabilistic thinking and actions, such as conjecturing, hypothesising, analysing, proving, evaluating, explaining, inferring, justifying, refuting, abstracting and generalising. Students are reasoning mathematically when they:

Information Communication Technologies (ICT) are powerful tools that can support student learning. Students can develop and demonstrate their understanding of concepts and content in Mathematics using a range of ICT tools. It is also important that students know how to use these ICT efficiently and responsibly, as well as learning how to protect themselves and secure their data.

HWB.GOV.WALES uses cookies which are essential for the site to work. Non-essential cookies are also used to tailor and improve services. By continuing to use this site, you agree to our use of cookies.

This provides specific guidance to be used when incorporating learning in mathematics and numeracy in your curriculum. It should be read together with the overarching Designing your Curriculum section which is relevant to learning and teaching across all areas of learning and experience.

A curriculum must embed the mandatory cross-curricular skills and the integral skills which underpin the four purposes of the curriculum. Below are some key principles which settings and schools should consider when designing learning and teaching in the Mathematics and Numeracy Area of Learning and Experience (Area).

Practitioners should develop engaging and accessible experiences where learners are given regular opportunities to describe, explain and justify their understanding of various mathematical concepts, using appropriate mathematical vocabulary. Literacy skills can also be developed in order to describe mathematical processes, such as reasoning, understanding a range of calculation strategies, describing visualisation of shapes, studying and interpreting information in statistics and comparing alternative methods before arriving at a solution to a mathematical problem. These literacy skills can be used as they encounter practical, real-life problems.

The development of logical and critical thinking underpins learning in mathematics. Mathematics teaches us problem-solving skills which transfer to all areas of the curriculum, to life in general and to the world of work. Mathematics involves solving problems and begins by analysing the requirements, before then asking questions and evaluating information. In the development of solutions, learners identify potential approaches and develop arguments, justifying their decisions.

Studying mathematics develops personal effectiveness. When studying mathematics everyone encounters challenges at some point, and overcoming these challenges requires and develops resourcefulness and resilience. Communicating about mathematical thinking and solving problems is a core aspect of mathematics. Mathematical communication is precise and logical and will be useful in life generally.

Mathematical thinking requires learners to be organised and, as they progress through school, their organisational skills will develop, particularly as they plan and implement the sequential data-handling cycle. In their mathematical problem-solving, learners should be encouraged to predict and estimate solutions and then to check their answers, reflect on their results and evaluate their approaches. Increasing confidence in decision-making for mathematical problem-solving supports learners to be more aspirational in setting goals and challenges for themselves including planning how to achieve these.

The different areas of mathematics are highly interconnected and dependent on one another and concepts are built up over time, drawing on prior knowledge and learning, often from more than one area of mathematics. What is important when planning to teach any specific topic is to be aware of the prior knowledge the learners need in order to access and understand the new topic.

Making connections between arithmetic and algebra helps to develop tools and skills for abstract reasoning from an early age. Measure is an aspect of geometrical thinking which is closely connected to number, and much of the development of understanding of number can emerge through increasingly sophisticated measuring. Geometric thinking involves proportional reasoning, which connects with development in number work; it also involves transforming shapes, which relates to the use of functions and mapping in algebra. Probability is expressed through number in various ways, using percentages, fractions and decimals, and an understanding of the different representations; the connections between them are necessary for effective expression of probability. Statistics involves manipulation, representation and interpretation of data, which in turn require numerical and geometric thinking.

Understanding that a half is a result of dividing something into two equal parts. This could be through connecting concrete and/or real-life experiences of partitioning objects and numbers into equal parts with images (e.g. pictures and images on the number line) and the abstract representation of a using the symbolic notation. A learner who understands what a half is might be able to give real-life or visual examples, and would also be able to explain why something might not be a half (e.g. a pizza cut into two parts which are not equal).

Being able to recognise real-life situations which involve a half and being able to represent these mathematically; being able to model situations involving halving mathematically; using pictures/images and language and symbols to illustrate a half.

Being able to understand the relationship between a half and a whole; being able to justify why is also a half. Being able to reason that + = 1, x 2 = 1 and 1 2 = . Being able to justify why there may be many ways of splitting a shape in half.

Numeracy and knowledge of real-world contexts could include understanding exchange rates, mortgage calculations and taxation, including the developing system of Welsh taxes. Through algebraic thinking and knowledge, learners could develop capabilities that can be applied to topics including personal finance and energy production in Wales and elsewhere. Geometry could draw examples from, and help develop a knowledge of, urban development as it differs across Wales, medical technology and computer imaging. From a strong foundation of problem solving, logical reasoning and understanding data, Mathematics and Numeracy is critical for informed citizens who are ready to play a full part in life and work.

3a8082e126