# Mathematics (2020)

## Cross-Curricular and Integrated Learning in Mathematics

When planning an integrated mathematics program, educators should consider that, although the mathematical content in the curriculum is outlined in discrete strands, students develop mathematical thinking, such as proportional reasoning, algebraic reasoning, and spatial reasoning, that transcends the expectations in the strands and even connects to the learning in many other subject areas. By purposefully drawing connections across all areas of mathematics and to other subject areas, and by applying learning to relevant real-life contexts, teachers extend and enhance student learning experiences and deepen their knowledge and skills across disciplines and beyond the classroom.

For example, proportional reasoning, which is developed through the study of ratios and rates in the Number strand, is also used when students are working towards meeting learning expectations in other strands of the math curriculum, such as Spatial Sense, and in other disciplines, such as science, geography, and the arts. Students then apply this learning in their everyday lives – for example, when adjusting a recipe or preparing a mixture or solutions.

Similarly, algebraic reasoning is applied beyond the Algebra strand. For example, it is applied in measurement when learning about formulas, such as *area of a parallelogram = base × height*. It is applied in other disciplines, such as science, when students study simple machines and learn about the formula *work* = *force* × *distance.* Algebraic reasoning is also used when making decisions in everyday life – for example, when determining which service provider offers a better consumer contract or when calculating how much time it will take for a frozen package to thaw.

Spatial thinking has a fundamental role throughout the Ontario curriculum, from Kindergarten to Grade 12, including in mathematics, the arts, health and physical education, and science. For example, a student demonstrates spatial reasoning when mentally rotating and matching shapes in mathematics, planning their move to the basketball hoop, and using diagonal and converging lines to create perspective drawing in visual art. In everyday life, there are many applications of spatial reasoning, such as when creating a garden layout or when using a map to navigate the most efficient way of getting from point A to point B.

Teaching mathematics as a narrowly defined subject area places limits on the depth of learning that can occur. When teachers work together to develop integrated learning opportunities and highlight cross-curricular connections, students are better able to:

- make connections between mathematics and other subject areas, and among the strands of the mathematics curriculum;
- improve their ability to provide multiple responses to a problem;
- debate and test whether responses are effective and efficient;
- apply a range of knowledge and skills to solve problems in mathematics and in their daily experiences and lives.

When students are provided with opportunities to learn mathematics through real-life applications, integrating learning expectations from across the curriculum, they use their knowledge of other subject matter to enhance their learning of and engagement in mathematics. More information about integrating learning across the curriculum can be found in “Cross-Curricular and Integrated Learning”.

### Literacy in Mathematics

Literacy skills needed for reading and writing in general are essential for the learning of mathematics. To engage in mathematical activities and develop computational fluency, students require the ability to read and write mathematical expressions, to use a variety of literacy strategies to comprehend mathematical text, to use language to analyse, summarize, and record their observations, and to explain their reasoning when solving problems. Research shows that “mathematics texts contain more concepts per sentence and paragraph than any other type of text”. Reading a mathematics text requires specific literacy strategies, unique to mathematics. Learning in “some areas of math in particular, such as word problems and number combinations may be mediated by language and reading due to the nature of the task.” As a result, there is a strong correlation between reading and math achievement.

The learning of mathematics requires students to navigate discipline-specific texts that “must be written and read in appropriate ways”; therefore, it is important that math instruction addresses both “mathematical texts and literacies.” Many of the activities and tasks students undertake in mathematics involve the use of written, oral, visual, and multimodal communication skills as they encounter mathematical texts such as “equations, graphs, diagrams, proofs, justifications, displays of manipulatives (e.g., base ten blocks), calculator readouts, verbal mathematical discussions, and written descriptions of problems.” The language of mathematics includes special terminology. To support all students in developing an understanding of mathematical texts, teachers need to explicitly teach mathematical vocabulary, focusing on the many meanings and applications of the terms students may encounter. In all mathematics programs, students are required to use appropriate and correct terminology and are encouraged to use language with care and precision in order to communicate effectively.

More information about the importance of literacy across the curriculum can be found in the “Literacy” and “Mathematical Literacy” subsections of “Cross-curricular and Integrated Learning”.

### Transferable Skills in Mathematics

The Ontario curriculum emphasizes a set of skills that are critical to all students’ ability to thrive in school, in the world beyond school, and in the future. These are known as transferable skills. Educators facilitate students’ development of transferable skills across the curriculum, from Kindergarten to Grade 12. They are as follows:

**Critical Thinking and Problem Solving.**In mathematics, students and educators learn and apply strategies to understand and solve problems flexibly, accurately, and efficiently. They learn to understand and visualize a situation and use the tools and language of mathematics to reason, make connections to real-life situations, communicate, and justify solutions.**Innovation, Creativity, and Entrepreneurship.**In mathematics, students and educators solve problems with curiosity, creativity, and a willingness to take risks. They pose questions, make and test conjectures, and consider problems from different perspectives to generate new learning and apply it to novel situations.**Self-Directed Learning.**By reflecting on their own thinking and emotions, students, with the support of educators, can develop perseverance, resourcefulness, resilience, and a sense of self. In mathematics, they initiate new learning, monitor their thinking and their emotions when solving problems and apply strategies to overcome challenges. They see mathematics as useful, interesting, and doable and confidently look for ways to apply their learning.**Collaboration.**In mathematics, students and educators engage with others productively, respectfully, and critically in order to better understand ideas and problems, generate solutions, and refine their thinking.**Communication.**In mathematics, students and educators use the tools and language of mathematics to describe their thinking and to understand the world. They use mathematical vocabulary, symbols, conventions, and representations to make meaning, express a point of view, and make convincing and compelling arguments in a variety of ways, including multimodally, for example, using combinations of oral, visual, textual, and gestural communication.**Global Citizenship and Sustainability.**In mathematics, students and educators recognize and appreciate multiple ways of knowing, doing, and learning, and value different perspectives. They see how mathematics is used in all walks of life and how engaged citizens can use it as a tool to raise awareness and generate solutions for real-life issues.**Digital Literacy.**In mathematics, students and educators learn to be discerning users of technology. They select when and how to use appropriate tools to understand and model real-life situations, predict outcomes, and solve problems, and they assess and evaluate the reasonableness of their results.

More information on instructional approaches can be found in the "Transferable Skills" section of "Program Planning".