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US Curriculum - CCSS |
Cross cutting themes - National Math Curriculum 2023 |
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Cross-cutting Themes |
A brief mention of the importance of mathematics in everyday life in the introduction but no SLOs connecting mathematical concepts to other disciplines or applications of mathematics |
A brief mention of the importance of mathematics in everyday life in the introduction but no SLOs connecting mathematical concepts to other disciplines or applications of mathematics |
Aims Section: The aims are to enable students to: • develop a positive attitude towards mathematics in a way that encourages enjoyment, establishes confidence and promotes enquiry and further learning • develop a feel for number and understand the significance of the results obtained • apply their mathematical knowledge and skills to their own lives and the world around them • use creativity and resilience to analyse and solve problems • communicate mathematics clearly • develop the ability to reason logically, make inferences and draw conclusions • develop fluency so that they can appreciate the interdependence of, and connections between, different areas of mathematics • acquire a foundation for further study in mathematics and other subjects. |
Aims Section: The aims are to enable students to: • develop their mathematical knowledge and skills in a way which encourages confidence and provides satisfaction and enjoyment • develop an understanding of mathematical principles and an appreciation of mathematics as a logical and coherent subject • acquire a range of mathematical skills, particularly those which will enable them to use applications of mathematics in the context of everyday situations and of other subjects they may be studying • develop the ability to analyse problems logically • recognise when and how a situation may be represented mathematically, identify and interpret relevant factors and select an appropriate mathematical method to solve the problem • use mathematics as a means of communication with emphasis on the use of clear expression • acquire the mathematical background necessary for further study in mathematics or related subjects. |
The aims of MYP mathematics are to encourage and enable students to: • enjoy mathematics, develop curiosity and begin to appreciate its elegance and power • develop an understanding of the principles and nature of mathematics • communicate clearly and confidently in a variety of contexts • develop logical, critical and creative thinking • develop confidence, perseverance, and independence in mathematical thinking and problem-solving • develop powers of generalization and abstraction • apply and transfer skills to a wide range of real-life situations, other areas of knowledge and future developments • appreciate how developments in technology and mathematics have influenced each other • appreciate the moral, social and ethical implications arising from the work of mathematicians and the applications of mathematics • appreciate the international dimension in mathematics through an awareness of the universality of mathematics and its multicultural and historical perspectives • appreciate the contribution of mathematics to other areas of knowledge • develop the knowledge, skills and attitudes necessary to pursue further studies in mathematics • develop the ability to reflect critically upon their own work and the work of others. |
enable students to: 1. develop a curiosity and enjoyment of mathematics, and appreciate its elegance and power 2. develop an understanding of the concepts, principles and nature of mathematics 3. communicate mathematics clearly, concisely and confidently in a variety of contexts 4. develop logical and creative thinking, and patience and persistence in problem solving to instil confidence in using mathematics 5. employ and refine their powers of abstraction and generalization 6. take action to apply and transfer skills to alternative situations, to other areas of knowledge and to future developments in their local and global communities 7. appreciate how developments in technology and mathematics influence each other 8. appreciate the moral, social and ethical questions arising from the work of mathematicians and the applications of mathematics 9. appreciate the universality of mathematics and its multicultural, international and historical perspectives 10. appreciate the contribution of mathematics to other disciplines, and as a particular “area of knowledge” in the TOK course 11. develop the ability to reflect critically upon their own work and the work of others 12. independently and collaboratively extend their understanding of mathematics. |
The CCSS states Mathematical practice standards that are to be included across the curriculum: 1. Make sense of problems & persevere in solving them 2. Reason abstractly & quantitatively 3. Construct viable arguments & critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for & make use of structure 8. Look for & express regularity in repeated reasoning |
Mathematics in the NCC curriculum is approached as an interdisciplinary subject. The idea of Science, Technology, Engineering, The Arts and Mathematics (STEAM) is used an overarching idea for how to break up the study of Mathematics into core disciplinary knowledge (the key concepts and practices of mathematics) and cross-cutting themes (the connections of Math with other disciplines).
Science, Technology and Engineering: applications of mathematics to create solutions that improve standards of living and the connections of mathematics with the natural world.
Arts: What can be understood about the nature of mathematics from the fine arts, performing arts and the humanities.
Mathematics: theoretical understandings/big ideas in mathematics and mathematical practices, and their mutual overlaps in the methods of mathematical inquiry. |
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Science, Technology and Engineering |
Mentioned in introduction: The curriculum recognizes the benefits that current technologies can bring to the learning and doing mathematics. It, therefore, integrates the use of appropriate technologies to enhance learning in an ever increasingly information-rich world. |
Mentioned in introduction: The curriculum recognizes the benefits that current technologies can bring to the learning and doing mathematics. It, therefore, integrates the use of appropriate technologies to enhance learning in an ever increasingly information-rich world.
Mentioned in Grade XI-XII SLOs: Integrate technology to aid the process of mathematical exploration. There is an entire unit on MAPLE and doing calculations using MAPLE commands have been integrated into the SLOs. |
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Global contexts: • identities and relationships • orientation in space and time • personal and cultural expression • scientific and technical innovation • globalization and sustainability • fairness and development. |
International mindedness: Science and technology are of significant importance in today’s world. As the language of science, mathematics is an essential component of most technological innovation and underpins developments in science and technology, although the contribution of mathematics may not always be visible. Examples of this include the role of the binary number system, matrix algebra, network theory and probability theory in the digital revolution, or the use of mathematical simulations to predict future climate change or spread of disease. These examples highlight the key role mathematics can play in transforming the world around us. |
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Applications of Mathematics 1. The interconnectedness of Mathematics and Science - The symbolic language of mathematics is extremely valuable for expressing scientific ideas unambiguously. - Mathematics provides the rules for analyzing scientific ideas and data rigorously. - The accuracy and reliability of mathematical theories and principles serve as a basis for scientific discovery and understanding. - Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty. - Science provides mathematics with interesting problems to investigate, and mathematics provides science with powerful tools to use in analyzing data.
2. Mathematics has a wide range of applications in science, engineering and technology. - Mathematics is often used as a tool in the sciences, such as physics, chemistry, and biology, to describe and explain phenomena in the natural world. - Mathematical models and equations are used to make predictions and test hypotheses in scientific research. - Engineers use mathematical concepts and techniques to solve practical problems and design systems and structures. - Engineers use mathematical models to simulate and analyze the behavior of systems, and to optimize their designs. Engineers also use mathematical tools to analyze and control complex systems and processes. - Mathematical methods and techniques are used to analyze and optimize the performance of a wide variety of technological systems and devices, including communication systems, control systems, and manufacturing processes. |
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Arts |
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Statements of inquiry: - Architects and engineers must use finite resources responsibly when they design new structures. - Logic is a powerful tool for justifying what we discover through measurement and observation. - Decision-making can be improved by using a model to represent relationships. - Understanding form and shape enhances creativity. - Modelling using a logical process helps us to understand the world. - Discovering mathematical relationships can lead to a better understanding of how environmental systems evolve. - Establishing patterns in the natural world can help in understanding relationships. |
International mindedness: - Many of the foundations of modern mathematics were laid many centuries ago by diverse civilisations – Arabic, Greek, Indian and Chinese among others. Mathematics can in some ways be seen as an international language and, apart from slightly differing notation, mathematicians from around the world can communicate effectively within their field. Mathematics can transcend politics, religion and nationality, and throughout history great civilizations have owed their success in part to their mathematicians being able to create and maintain complex social and architectural structures.
As part of their theory of knowledge course, students are encouraged to explore tensions relating to knowledge in mathematics. As an area of knowledge, mathematics seems to supply a certainty perhaps impossible in other disciplines and in many instances provides us with tools to debate these certainties. This may be related to the “purity” of the subject, something that can sometimes make it seem divorced from reality. Yet mathematics has also provided important knowledge about the world and the use of mathematics in science and technology has been one of the driving forces for scientific advances. Despite all its undoubted power for understanding and change, mathematics is in the end a puzzling phenomenon. A fundamental question for all knowers is whether mathematical knowledge really exists independently of our thinking about it. Is it there, “waiting to be discovered”, or is it a human creation? Indeed, the philosophy of mathematics is an area of study in its own right. |
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Nature of Mathematics 1. Mathematics is a product of the exploration of structure, patterns and relationships. - As a theoretical discipline, mathematics is driven by abstract concepts and generalization. This mathematics is drawn out of ideas, and develops through linking these ideas and developing new ones. - As an applied discipline, mathematicians focus their attention on solving problems and discovering relationships that originate in the world of experience. - The results of theoretical and applied mathematics often influence each other.
2. Mathematics uses a variety of methods to make claims. - Mathematics uses multiple strategies and multiple representations to revise and produce new knowledge. - The new knowledge is presented in the form of theorems that have been built from axioms and logical mathematical arguments and a theorem is only accepted as true when it has been proven. - Mathematics relies on logic rather than on observation as its standard of validity and accuracy, yet employs observation, simulation, and even experimentation as means of discovering new ideas, theories and principles.
3. Mathematical knowledge is open to revision and refinement. - Mathematics has a history that includes the refinement of, and changes to, theories, ideas, and beliefs over time. - Mathematics is critiqued and verified by people within particular cultures through justification or proof that is communicated to oneself and others. - The body of knowledge that makes up mathematics is not fixed; it has grown during human history and is growing at an increasing rate.
4. Mathematics is a Human Endeavor. - Mathematical knowledge is a result of human endeavor, imagination and creativity. - Mathematics can be produced by each and every person. - Mathematics is not created arbitrarily, but arises from activity with already existing mathematical objects, and from the needs of science and daily life. - Individuals and teams from many nations and cultures have contributed to mathematics and to advances in mathematical applications in science, engineering and technology. - Mathematicians’ backgrounds, theoretical commitments, and fields of endeavor influence the nature of their findings. - Technological advances have influenced the progress of mathematics and mathematics has influenced advances in technology. - Mathematical ideas impact society and culture, and cultural and societal factors influence the development of mathematics.
5. Mathematics is worthwhile, beautiful and often useful. - Mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems. - Mathematics empowers us to better understand the information-laden world in which we live by equipping us with critical thinking skills. - Mathematics reveals hidden patterns that help us understand the natural world around us. - The patterns and structures that exist in mathematics are considered to be aesthetically pleasing and beautiful, much like works of art. - Mathematics is a language that is understood and used globally, making it a bridge between cultures and disciplines. |
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Mathematics |
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The key concepts for Cambridge International AS & A Level Mathematics are: • Problem solving Mathematics is fundamentally problem solving and representing systems and models in different ways. These include: – Algebra: this is an essential tool which supports and expresses mathematical reasoning and provides a means to generalise across a number of contexts. – Geometrical techniques: algebraic representations also describe a spatial relationship, which gives us a new way to understand a situation. – Calculus: this is a fundamental element whichdescribes change in dynamic situations and underlines the links between functions and graphs. – Mechanical models: these explain and predict how particles and objects move or remain stable under the influence of forces. – Statistical methods: these are used to quantify and model aspects of the world around us. Probability theory predicts how chance events might proceed, and whether assumptions about chance are justified by evidence. • Communication Mathematical proof and reasoning is expressed using algebra and notation so that others can follow each line of reasoning and confirm its completeness and accuracy. Mathematical notation is universal. Each solution is structured, but proof and problem solving also invite creative and original thinking. • Mathematical modelling Mathematical modelling can be applied to many different situations and problems, leading to predictions and solutions. A variety of mathematical content areas and techniques may be required to create the model. Once the model has been created and applied, the results can be interpreted to give predictions and information about the real world. |
Key concepts:
1. Form Form is the shape and underlying structure of an entity or piece of work, including its organization, essential nature and external appearance. Form in MYP mathematics refers to the understanding that the underlying structure and shape of an entity is distinguished by its properties. Form provides opportunities for students to appreciate the aesthetic nature of the constructs used in a discipline.
2. Logic Logic is a method of reasoning and a system of principles used to build arguments and reach conclusions. Logic in MYP mathematics is used as a process in making decisions about numbers, shapes, and variables. This system of reasoning provides students with a method for explaining the validity of their conclusions. Within the MYP, this should not be confused with the subfield of mathematics called “symbolic logic”.
3. Relationships Relationships are the connections and associations between properties, objects, people and ideas— including the human community’s connections with the world in which we live. Any change in relationship brings consequences—some of which may occur on a small scale, while others may be far reaching, affecting large networks and systems such as human societies and the planetary ecosystem. Relationships in MYP mathematics refers to the connections between quantities, properties or concepts and these connections may be expressed as models, rules or statements. Relationships provide opportunities for students to explore patterns in the world around them. Connections between the student and mathematics in the real world are important in developing deeper understanding.
Related concepts: Change Equivalence Generalization Justification Measurement Models Patterns Quantity Representation Simplification Space Systems |
KNOWLEDGE Approximation: This concept refers to a quantity or a representation which is nearly but not exactly correct. Change: This concept refers to a variation in size, amount or behaviour. Equivalence: This concept refers to the state of being identically equal or interchangeable, applied to statements, quantities or expressions. Generalization: This concept refers to a general statement made on the basis of specific examples. Modelling: This concept refers to the way in which mathematics can be used to represent the real world. Patterns: This concept refers to the underlying order, regularity or predictability of the elements of a mathematical system. Quantity: This concept refers to an amount or number. Relationships: This concept refers to the connection between quantities, properties or concepts; these connections may be expressed as models, rules or statements. Relationships provide opportunities for students to explore patterns in the world around them. Representation: This concept refers to using words, formulae, diagrams, tables, charts, graphs and models to represent mathematical information. Space: This concept refers to the frame of geometrical dimensions describing an entity. Systems: This concept refers to groups of interrelated elements. Validity: This concept refers to using well-founded, logical mathematics to come to a true and accurate conclusion or a reasonable interpretation of results.
PRACTICES Mathematical Inquiry Modeling Proof Use of technology |
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A. Mathematical Knowledge (these themes represent big ideas in mathematics which are applied across the conceptual SLOs)
1. Quantity, Measurement and Approximation - Quantities and values can be used to describe key features and behaviours of objects such as functions. - Measurements can be represented in equivalent ways using different units. For example, degrees and radians can be used for angles to facilitate ease of calculation. - Approximation of numbers adds uncertainty or inaccuracy to calculations, leading to potential errors but can be useful when handling extremely large or small quantities. - When quantities change, a useful measurement to make is the “Rate of Change” which gives us an idea of how much one quantity is dependent on the other.
2. Abstraction and generalization - Mathematical situations and structures can be translated and represented abstractly using variables, expressions, and equations. - Extending results from a specific case to a general form can allow us to apply them to a larger system.
3. Patterns, relationships and modelling systems - Patterns can be identified in behaviours which can give us insight into appropriate strategies to model or solve them. - Relationships can be described and generalizations made for mathematical situations that have numbers or objects that repeat in predictable ways. - Modelling real-life situations allows for prediction, analysis and interpretation and can be used to provide effective solutions to real-life problems. - Predictions based on models have limited precision and reliability due to the assumptions and approximations inherent in models.
4. Representation and Equivalence - Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value. - Different but equivalent representations of objects such as visual, symbolic, verbal, contextual and physical representations, can reveal different characteristics of the same relationship. - Different representations enable quantities to be compared and used for computational purposes with ease and accuracy.
5. Space - Objects in space can be oriented in an infinite number of ways, and an object’s location in space can be described quantitatively. - Objects in space can be transformed in an infinite number of ways, and those transformations can be described and analyzed mathematically.
6. Logic, validity and justification - Logic is a powerful tool for justifying what we discover through measurement and observation. - Logic is a method of reasoning and a system of principles used to build arguments, reach conclusions and explain the validity of these conclusions. - Considering the reasonableness and validity of results helps us to make informed, unbiased decisions.
B. Mathematical Practices (these themes are also embedded in the conceptual SLOs but will primarily be implemented through teaching and learning practices elaborated in the teacher guide)
1. Problem-solving - Understand the meaning of a problem and look for entry points to its solution. - Analyze givens, constraints, relationships, and goals. - Make conjectures about the form and meaning of the solution and plan a solution pathway. - Employ different problem solving strategies in order to gain insight into its solution. These can include: • Considering analogous problems • Trying special cases and simpler forms of the original problem • Finding patterns or structure and looking for general methods • Listing all possibilities and eliminating options based on constraints • Making educated guesses and using trial and error • Visualizing the problem using different diagrams • Working backwards - Monitor and evaluate progress and check answers to problems using a different method. - Understand the approaches of others to solving complex problems and identify correspondences between different approaches.
2. Communication and reasoning - While constructing arguments, understand and use stated assumptions, clear definitions, and previously established results, considering the units involved and attending to the meaning of quantities and symbols. - Make conjectures and build a logical progression of statements to explore the truth of the conjectures. - Analyze situations by breaking them into cases, and recognize and use counterexamples. - Justify conclusions, communicate them to others, and respond to the arguments of others. - Ask useful questions to clarify or improve the arguments. - Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and explain the flaw in an incorrect argument.
3. Mathematical modelling - Apply mathematical knowledge to solve problems arising in everyday life, society, and the workplace. - Make choices, assumptions and approximations to simplify a complicated situation. - Identify variables in the situation and select those that represent essential features. - Formulate a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables. - Analyze these relationships mathematically to draw conclusions. - Interpret the mathematical results in the context of the original situation. - Validate the conclusions by comparing them with the situation, and improve the model if it has not served its purpose.
4. Use appropriate tools strategically - Able to use tools, including technological tools, to explore and deepen their understanding of concepts, solve mathematical problems, test conjectures and justify interpretations. - Be familiar with the different kinds of non-technological tools available such as pencil and paper, concrete models, ruler, protractor and calculator. - Be familiar with the different kinds of technological tools available such as graphical calculators, dynamic graphing software, spreadsheets, simulations, apps, and dynamic geometry software. - Make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. |
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