Students learn mathematics through the experiences that teachers provide. Teachers must know and understand deeply the mathematics they are teaching and understand and be committed to their students as learners of mathematics and as human beings. There is no one “right way” to teach. Nevertheless, much is known about effective mathematics teaching. Selecting and using suitable curricular materials, using appropriate instructional tools and techniques to support learning, and pursuing continuous self-improvement are actions good teachers take every day. The teacher is responsible for creating an intellectual environment in the classroom where serious engagement in mathematical thinking is the norm.
Effective teaching requires deciding what aspects of a task to highlight, how to organize and orchestrate the work of students, what questions to ask students having varied levels of expertise, and how to support students without taking over the process of thinking for them. Effective teaching requires continuing efforts to learn and improve.
Teachers need to increase their knowledge about mathematics and pedagogy, learn from their students and colleagues, and engage in professional development and self-reflection. Collaborating with others–pairing an experienced teacher with a new teacher or forming a community of teachers–to observe, analyze, and discuss teaching and students’ thinking is a powerful, yet neglected, form of professional development. Teachers need ample opportunities to engage in this kind of continual learning. The working lives of teachers must be structured to allow and support different models of professional development that benefit them and their students.
Principles and practice
What can learning in mathematics enable children and young people to achieve?
Mathematics is important in our everyday life, allowing us to make sense of the world around us and to manage our lives. Using mathematics enables us to model real-life situations and make connections and informed predictions. It equips us with the skills we need to interpret and analyse information, simplify and solve problems, assess risk and make informed decisions.
Mathematics plays an important role in areas such as science or technologies, and is vital to research and development in fields such as engineering, computing science, medicine and finance. Learning mathematics gives children and young people access to the wider curriculum and the opportunity to pursue further studies and interests.
Because mathematics is rich and stimulating, it engages and fascinates learners of all ages, interests and abilities. Learning mathematics develops logical reasoning, analysis, problem-solving skills, creativity and the ability to think in abstract ways. It uses a universal language of numbers and symbols which allows us to communicate ideas in a concise, unambiguous and rigorous way.
To face the challenges of the 21st century, each young person needs to have confidence in using mathematical skills, and Scotland needs both specialist mathematicians and a highly numerate population.
Building the Curriculum 1
Mathematics equips us with many of the skills required for life, learning and work. Understanding the part that mathematics plays in almost all aspects of life is crucial. This reinforces the need for mathematics to play an integral part in lifelong learning and be appreciated for the richness it brings.
How is the mathematics framework structured?
Within the mathematics framework, some statements of experiences and outcomes are also identified as statements of experiences and outcomes in numeracy. These form an important part of the mathematics education of all children and young people as they include many of the numerical and analytical skills required by each of us to function effectively and successfully in everyday life. All teachers with a responsibility for the development of mathematics will be familiar with the role of numeracy within mathematics and with the means by which numeracy is developed across the range of learning experiences. The numeracy subset of the mathematics experiences and outcomes is also published separately; further information can be found in the numeracy principles and practice paper.
The mathematics experiences and outcomes are structured within three main organisers, each of which contains a number of subdivisions:
Number, money and measure
Estimation and rounding
Number and number processes
Multiples, factors and primes
Powers and roots
Fractions, decimal fractions and percentages
Mathematics – its impact on the world, past, present and future
Patterns and relationships
Expressions and equations.
Shape, position and movement
Properties of 2D shapes and 3D objects
Angle, symmetry and transformation.
Data and analysis
Ideas of chance and uncertainty.
The mathematics framework as a whole includes a strong emphasis on the important part mathematics has played, and will continue to play, in the advancement of society, and the relevance it has for daily life.
A key feature of the mathematics framework is the development of algebraic thinking from an early stage. Research shows that the earlier algebraic thinking is introduced, the deeper the mathematical understanding will be and the greater the confidence in using mathematics.
Teachers will use the statements of experiences and outcomes in information handling to emphasise the interpretation of statistical information in the world around us and to emphasise the knowledge and skills required to take account of chance and uncertainty when making decisions.
The level of achievement at the fourth level has been designed to approximate to that associated with SCQF level 4.
What are the features of effective learning and teaching in mathematics?
From the early stages onwards, children and young people should experience success in mathematics and develop the confidence to take risks, ask questions and explore alternative solutions without fear of being wrong. They will enjoy exploring and applying mathematical concepts to understand and solve problems, explaining their thinking and presenting their solutions to others in a variety of ways. At all stages, an emphasis on collaborative learning will encourage children to reason logically and creatively through discussion of mathematical ideas and concepts.
Through their use of effective questioning and discussion, teachers will use misconceptions and wrong answers as opportunities to improve and deepen children’s understanding of mathematical concepts.
The experiences and outcomes encourage learning and teaching approaches that challenge and stimulate children and young people and promote their enjoyment of mathematics. To achieve this, teachers will use a skilful mix of approaches, including: planned active learning which provides opportunities to observe, explore, investigate, experiment, play, discuss and reflect modelling and scaffolding the development of mathematical thinking skills learning collaboratively and independently opportunities for discussion, communication and explanation of thinking developing mental agility using relevant contexts and experiences, familiar to young people making links across the curriculum to show how mathematical concepts are applied in a wide range of contexts, such as those provided by science and social studies using technology in appropriate and effective ways building on the principles of Assessment is for Learning, ensuring that young people understand the purpose and relevance of what they are learning developing problem-solving capabilities and critical thinking skills.
Mathematics is at its most powerful when the knowledge and understanding that have been developed are used to solve problems. Problem solving will be at the heart of all our learning and teaching. We should regularly encourage children and young people to explore different options: ‘what would happen if…?’ is the fundamental question for teachers and learners to ask as mathematical thinking develops.
How will we ensure progression within and through levels?
As children and young people develop concepts within mathematics, these will need continual reinforcement and revisiting in order to maintain progression. Teachers can plan this development and progression through providing children and young people with more challenging contexts in which to use their skills. When the experience or outcome spans two levels within a line of development, this will be all the more important.
One case in point would be the third level outcome on displaying information. The expectation is that young people will continue to use and refine the skills developed at second level to display charts, graphs and diagrams. The contexts should ensure progression and there are clear opportunities to use other curriculum areas when extending young people’s understanding.
What are broad features of assessment in mathematics?
(This section should be read alongside the advice for numeracy.)
Assessment in mathematics will focus on children and young people’s abilities to work increasingly skilfully with numbers, data and mathematical concepts and processes and use them in a range of contexts. Teachers can gather evidence of progress as part of day-to-day learning about number, money and measurement, shape, position and movement and information handling. The use of specific assessment tasks will be important in assessing progress at key points of learning including transitions.
From the early years through to the senior stages, children and young people will demonstrate progress in their skills in interpreting and analysing information, simplifying and solving problems, assessing risk and making informed choices. They will also show evidence of progress through their skills in collaborating and working independently as they observe, explore, experiment with and investigate mathematical problems.
Approaches to assessment should identify the extent to which children and young people can apply their skills in their learning, in their daily lives and in preparing for the world of work. Progress will be seen as children and young people demonstrate their competence and confidence in applying mathematical concepts and skills. For example:
Do they relish the challenge of number puzzles, patterns and relationships? Can they explain increasingly more abstract ideas of algebraic thinking? Can they successfully carry out mathematical processes and use their developing range of skills and attributes as set out in the experiences and outcomes? As they apply these to problems, can they draw on skills and concepts learned previously? As they tackle problems in unfamiliar contexts, can they confidently identify which skills and concepts are relevant to the problem? Can they then apply their skills accurately and then evaluate their solutions? Can they explain their thinking and demonstrate their understanding of 2D shapes and 3D objects? Can they evaluate data to make informed decisions?
Are they developing the capacity to engage with and complete tasks and assignments? Assessment should also link with other areas of the curriculum, within and outside the classroom, offering children and young people opportunities to develop and demonstrate their understanding of mathematics through social studies, technologies and science, and cultural and enterprise activities.
How can I make connections within and beyond mathematics?
Within mathematics there are rich opportunities for links among different concepts: a ready example is provided by investigations into area and perimeter which can involve estimation, patterns and relationships and a variety of numbers. When children and young people investigate number processes, there will be regular opportunities to develop mental strategies and mental agility. Teachers will make use of opportunities to develop algebraic thinking and introduce symbols, such as those opportunities afforded at early stages when reinforcing number bonds or later when investigating the sum of the angles in a triangle.
There are many opportunities to develop mathematical concepts in all other areas of the curriculum. Patterns and symmetry are fundamental to art and music; time, money and measure regularly occur in modern languages, home economics, design technology and various aspects of health and wellbeing; graphs and charts are regularly used in science and social studies; scale and proportion can be developed within social studies; formulae are used in areas including health and wellbeing, technologies and sciences; while shape, position and movement can be developed in all areas of the curriculum.
The Teaching Principle
Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. Students learn mathematics through the experiences that teachers provide. Thus, students’ understanding of mathematics, their ability to » use it to solve problems, and their confidence in, and disposition toward, mathematics are all shaped by the teaching they encounter in school. The improvement of mathematics education for all students requires effective mathematics teaching in all classrooms. Teaching mathematics well is a complex endeavor, and there are no easy recipes for helping all students learn or for helping all teachers become effective. Nevertheless, much is known about effective mathematics teaching, and this knowledge should guide professional judgment and activity. To be effective, teachers must know and understand deeply the mathematics they are teaching and be able to draw on that knowledge with flexibility in their teaching tasks.
They need to understand and be committed to their students as learners of mathematics and as human beings and be skillful in choosing from and using a variety of pedagogical and assessment strategies (National Commission on Teaching and America’s Future 1996). In addition, effective teaching requires reflection and continual efforts to seek improvement. Teachers must have frequent and ample opportunities and resources to enhance and refresh their knowledge. Effective teaching requires knowing and understanding mathematics, students as learners, and pedagogical strategies. Teachers need several different kinds of mathematical knowledge—knowledge about the whole domain; deep, flexible knowledge about curriculum goals and about the important ideas that are central to their grade level; knowledge about the challenges students are likely to encounter in learning these ideas; knowledge about how the ideas can be represented to teach them effectively; and knowledge about how students’ understanding can be assessed.
This knowledge helps teachers make curricular judgments, respond to students’ questions, and look ahead to where concepts are leading and plan accordingly. Pedagogical knowledge, much of which is acquired and shaped through the practice of teaching, helps teachers understand how students learn mathematics, become facile with a range of different teaching techniques and instructional materials, and organize and manage the classroom. Teachers need to understand the big ideas of mathematics and be able to represent mathematics as a coherent and connected enterprise (Schifter 1999; Ma 1999). Their decisions and their actions in the classroom—all of which affect how well their students learn mathematics—should be based on this knowledge. This kind of knowledge is beyond what most teachers experience in standard preservice mathematics courses in the United States. For example, that fractions can be understood as parts of a whole, the quotient of two integers, or a number on a line is important for mathematics teachers (Ball and Bass forthcoming). Such understanding might be characterized as “profound understanding of fundamental mathematics” (Ma 1999).
Teachers also need to understand the different representations of an idea, the relative strengths and weaknesses of each, and how they are related to one another (Wilson, Shulman, and Richert 1987). They need to know the ideas with which students often have difficulty and ways to help bridge common misunderstandings. » Effective mathematics teaching requires a serious commitment to the development of students’ understanding of mathematics. Because students learn by connecting new ideas to prior knowledge, teachers must understand what their students already know. Effective teachers know how to ask questions and plan lessons that reveal students’ prior knowledge; they can then design experiences and lessons that respond to, and build on, this knowledge.
Teachers have different styles and strategies for helping students learn particular mathematical ideas, and there is no one “right way” to teach. However, effective teachers recognize that the decisions they make shape students’ mathematical dispositions and can create rich settings for learning. Selecting and using suitable curricular materials, using appropriate instructional tools and techniques, and engaging in reflective practice and continuous self-improvement are actions good teachers take every day. One of the complexities of mathematics teaching is that it must balance purposeful, planned classroom lessons with the ongoing decision making that inevitably occurs as teachers and students encounter unanticipated discoveries or difficulties that lead them into uncharted territory. Teaching mathematics well involves creating, enriching, maintaining, and adapting instruction to move toward mathematical goals, capture and sustain interest, and engage students in building mathematical understanding.
Effective teaching requires a challenging and supportive classroom learning environment. Teachers make many choices each day about how the learning environment will be structured and what mathematics will be emphasized. These decisions determine, to a large extent, what students learn. Effective teaching conveys a belief that each student can and is expected to understand mathematics and that each will be supported in his or her efforts to accomplish this goal. Teachers establish and nurture an environment conducive to learning mathematics through the decisions they make, the conversations they orchestrate, and the physical setting they create. Teachers’ actions are what encourage students to think, question, solve problems, and discuss their ideas, strategies, and solutions. The teacher is responsible for creating an intellectual environment where serious mathematical thinking is the norm. More than just a physical setting with desks, bulletin boards, and posters, the classroom environment communicates subtle messages about what is valued in learning and doing mathematics.
Are students’ discussion and collaboration encouraged? Are students expected to justify their thinking? If students are to learn to make conjectures, experiment with various approaches to solving problems, construct mathematical arguments and respond to others’ arguments, then creating an environment that fosters these kinds of activities is essential. In effective teaching, worthwhile mathematical tasks are used to introduce important mathematical ideas and to engage and challenge students intellectually. Well-chosen tasks can pique students’ curiosity and draw them into mathematics. The tasks may be connected to the » real-world experiences of students, or they may arise in contexts that are purely mathematical.
Regardless of the context, worthwhile tasks should be intriguing, with a level of challenge that invites speculation and hard work. Such tasks often can be approached in more than one way, such as using an arithmetic counting approach, drawing a geometric diagram and enumerating possibilities, or using algebraic equations, which makes the tasks accessible to students with varied prior knowledge and experience. Worthwhile tasks alone are not sufficient for effective teaching. Teachers must also decide what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge.
Opportunities to reflect on and refine instructional practice—during class and outside class, alone and with others—are crucial in the vision of school mathematics outlined in Principles and Standards. To improve their mathematics instruction, teachers must be able to analyze what they and their students are doing and consider how those actions are affecting students’ learning. Using a variety of strategies, teachers should monitor students’ capacity and inclination to analyze situations, frame and solve problems, and make sense of mathematical concepts and procedures. They can use this information to assess their students’ progress and to appraise how well the mathematical tasks, student discourse, and classroom environment are interacting to foster students’ learning.
They then use these appraisals to adapt their instruction. Reflection and analysis are often individual activities, but they can be greatly enhanced by teaming with an experienced and respected colleague, a new teacher, or a community of teachers. Collaborating with colleagues regularly to observe, analyze, and discuss teaching and students’ thinking or to do “lesson study” is a powerful, yet neglected, form of professional development in American schools (Stigler and Hiebert 1999). The work and time of teachers must be structured to allow and support professional development that will benefit them and their students.