The Mathematical Processes
Students learn and apply the mathematical processes as they work to achieve the expectations outlined in the curriculum. All students are actively engaged in applying these processes throughout the program. They apply these processes, together with social-emotional learning (SEL) skills, across the curriculum to support learning in mathematics. See the section “Strand A: Social-Emotional Learning (SEL) Skills in Mathematics and the Mathematical Processes” for more information.
The mathematical processes that support effective learning in mathematics are as follows:
- problem solving
- reasoning and proving
- selecting tools and strategies
The mathematical processes can be seen as the processes through which all students acquire and apply mathematical knowledge, concepts, and skills. These processes are interconnected. Problem solving and communicating have strong links to all the other processes. A problem-solving approach encourages students to reason their way to a solution or a new understanding. As students engage in reasoning, teachers further encourage them to pose questions, make conjectures, and justify solutions, orally and in writing. The communication and reflection that occur before, during, and after the process of problem solving help students not only to articulate and refine their thinking but also to see the problem they are solving from different perspectives. This opens the door to recognizing the range of strategies that can be used to arrive at a solution. By seeing how others solve a problem, students can begin to reflect on their own thinking (a process known as “metacognition”) and the thinking of others, as well as their own language use (a process known as “metalinguistic awareness”), and to consciously adjust their own strategies in order to make their solutions as efficient and accurate as possible.
The mathematical processes cannot be separated from the knowledge, concepts, and skills that students acquire throughout the year. All students problem solve, communicate, reason, reflect, and so on, as they develop the knowledge, the understanding of mathematical concepts, and the skills required in all the strands in every grade.
Problem solving is central to doing mathematics. By learning to solve problems and by learning through problem solving, students are given, and create, numerous opportunities to connect mathematical ideas and to develop conceptual understanding. Problem solving forms the basis of effective mathematics programs that place all students’ experiences and queries at the centre. Thus, problem solving should be the mainstay of mathematical instruction. It is considered an essential process through which all students are able to achieve the expectations in mathematics and is an integral part of the Ontario mathematics curriculum.
- increases opportunities for the use of critical thinking skills (e.g., selecting appropriate tools and strategies, estimating, evaluating, classifying, assuming, recognizing relationships, conjecturing, posing questions, offering opinions with reasons, making judgements) to develop mathematical reasoning;
- helps all students develop a positive math identity;
- allows all students to use the rich prior mathematical knowledge they bring to school;
- helps all students make connections among mathematical knowledge, concepts, and skills, and between the classroom and situations outside the classroom;
- promotes the collaborative sharing of ideas and strategies and promotes talking about mathematics;
- facilitates the use of creative-thinking skills when developing solutions and approaches;
- helps students find enjoyment in mathematics and become more confident in their ability to do mathematics.
Most importantly, when problem solving is in a mathematical context relevant to students’ experiences and derived from their own problem posing, it furthers their understanding of mathematics and develops their math agency.
Problem-Solving Strategies. Problem-solving strategies are methods that can be used to solve problems of various types. Common problem-solving strategies include the following: simulating; making a model, picture, or diagram; looking for a pattern; guessing and checking; making an organized list; making a table or chart; solving a simpler version of the problem (e.g., with smaller numbers); working backwards; and using logical reasoning. Teachers can support all students as they develop their use of these strategies by engaging with solving various kinds of problems – instructional problems, routine problems, and non-routine problems. As students develop this repertoire over time, they become more confident in posing their own questions, more mature in their problem-solving skills, and more flexible in using appropriate strategies when faced with new problem-solving situations.
Reasoning and Proving
Reasoning and proving are a mainstay of mathematics and involves students using their understanding of mathematical knowledge, concepts, and skills to justify their thinking. Proportional reasoning, algebraic reasoning, spatial reasoning, statistical reasoning, and probabilistic reasoning are all forms of mathematical reasoning. Students also use their understanding of numbers and operations, geometric properties, and measurement relationships to reason through solutions to problems. Teachers can provide all students with learning opportunities where they must form mathematical conjectures and then test or prove them to see if they hold true. Initially, students may rely on the viewpoints of others to justify a choice or an approach to a solution. As they develop their own reasoning skills, they will begin to justify or prove their solutions by providing evidence.
Students reflect when they are working through a problem to monitor their thought process, to identify what is working and what is not working, and to consider whether their approach is appropriate or whether there may be a better approach. Students also reflect after they have solved a problem by considering the reasonableness of their answer and whether adjustments need to be made. Teachers can support all students as they develop their reflecting and metacognitive skills by asking questions that have them examine their thought processes, as well as questions that have them think about other students’ thought processes. Students can also reflect on how their new knowledge can be applied to past and future problems in mathematics.
Experiences that allow all students to make connections – to see, for example, how knowledge, concepts, and skills from one strand of mathematics are related to those from another – will help them to grasp general mathematical principles. Through making connections, students learn that mathematics is more than a series of isolated skills and concepts and that they can use their learning in one area of mathematics to understand another. Seeing the relationships among procedures and concepts also helps develop mathematical understanding. The more connections students make, the deeper their understanding, and understanding, in turn, helps them to develop their sense of identity. In addition, making connections between the mathematics they learn at school and its applications in their everyday lives not only helps students understand mathematics but also allows them to understand how useful and relevant it is in the world beyond the classroom. These kinds of connections will also contribute to building students’ mathematical identities.
Communication is an essential process in learning mathematics. Students communicate for various purposes and for different audiences, such as the teacher, a peer, a group of students, the whole class, a community member, or their family. They may use oral, visual, written, or gestural communication. Communication also involves active and respectful listening. Teachers provide differentiated opportunities for all students to acquire the language of mathematics, developing their communication skills, which include expressing, understanding, and using appropriate mathematical terminology, symbols, conventions, and models.
For example, teachers can ask students to:
- share and clarify their ideas, understandings, and solutions;
- create and defend mathematical arguments;
- provide meaningful descriptive feedback to peers; and
- pose and ask relevant questions.
Effective classroom communication requires a supportive, safe, and respectful environment in which all members of the class feel comfortable and valued when they speak and when they question, react to, and elaborate on the statements of their peers and the teacher.
Students represent mathematical ideas and relationships and model situations using tools, pictures, diagrams, graphs, tables, numbers, words, and symbols. Teachers recognize and value the varied representations students begin learning with, as each student may have different prior access to and experiences with mathematics. While encouraging student engagement and affirming the validity of their representations, teachers help students reflect on the appropriateness of their representations and refine them. Teachers support students as they make connections among various representations that are relevant to both the student and the audience they are communicating with, so that all students can develop a deeper understanding of mathematical concepts and relationships. All students are supported as they use the different representations appropriately and as needed to model situations, solve problems, and communicate their thinking.
Selecting Tools and Strategies
Students develop the ability to select appropriate technologies, tools, and strategies to perform particular mathematical tasks, to investigate mathematical ideas, and to solve problems.
Technology. A wide range of technological and digital tools can be used in many contexts for students to interact with, learn, and do mathematics.
Students can use:
- calculators and computers to perform complex operations; create graphs; and collect, organize, and display data;
- digital tools, apps, and social media to investigate mathematical concepts and develop an understanding of mathematical relationships;
- statistical software to manipulate, analyse, represent, sort, and communicate data;
- software to code;
- dynamic geometry software and online geometry tools to develop spatial sense;
- computer programs to represent and simulate mathematical situations (i.e., mathematical modelling);
- communications technologies to support and communicate their thinking and learning;
- computers, tablets, and mobile devices to access mathematical information available on the websites of organizations around the world and to develop information literacy.
Developing the ability to perform mental computations is an important aspect of student learning in mathematics. Students must, therefore, use technology with discretion, and only when it makes sense to do so. When students use technology in their mathematics learning, they should apply mental computation, reasoning, and estimation skills to predict and check answers.
Tools. All students should be encouraged to select and use tools to illustrate mathematical ideas. Students come to understand that making their own representations is a powerful means of building understanding and of explaining their thinking to others. Using tools helps students:
- see patterns and relationships;
- make connections between mathematical concepts and between concrete and abstract representations;
- test, revise, and confirm their reasoning;
- remember how they solved a problem;
- communicate their reasoning to others, including by gesturing.
Strategies. Problem solving often requires students to select an appropriate strategy. Students learn to judge when an exact answer is needed and when an estimate is all that is required, and they use this knowledge to guide their selection. For example, computational strategies include mental computation and estimation to develop a sense of the numbers and operations involved. The selection of a computational strategy is based on the flexibility students have with applying operations to the numbers they are working with. Sometimes, their strategy may involve the use of algorithms or the composition and decomposition of numbers using known facts. Students can also create computational representations of mathematical situations using code.