Some Considerations for Program Planning in Mathematics
Teachers consider many factors when planning a mathematics program that cultivates the best possible environment in which all students can maximize their mathematical learning. This section highlights the key strategies and approaches that teachers and school leaders should consider as they plan effective and inclusive mathematics programs. Additional information can be found in the “Considerations for Program Planning” section, which provides information applicable to all curricula.
Instruction in mathematics should support all students in acquiring the knowledge, skills, and habits of mind they need in order to achieve the curriculum expectations and be able to enjoy and participate in mathematics learning for years to come.
Effective math instruction begins with knowing the identity and profile of the students, having high expectations for and of all students, and believing that all students can learn and do mathematics. It uses culturally relevant and responsive practices and differentiated learning experiences to meet individual students' learning strengths and needs. It focuses on the development of conceptual understanding and procedural fluency, skill development, communication, and problem-solving skills. It takes place in a safe, positive, and inclusive learning environment, where all students feel valued, empowered, engaged, and able to take risks and approach the learning of mathematics in a confident manner. Instruction that is student-centred and that builds on students’ strengths is effective as it motivates and engages students meaningfully and instils positive habits of mind, such as curiosity and open-mindedness; a willingness to think, to question, to challenge and be challenged; and an awareness of the value of listening intently, reading thoughtfully, and communicating with clarity.
Learning should be relevant and inspired by the lived realities of all students and embedded in authentic, real-life contexts that allow students to develop the fundamental mathematical concepts and skills and to see the beauty and wide-ranging nature of mathematics. This approach enables students to use mathematical reasoning to see connections throughout their lives.
Teachers understand the importance of knowing the identities and profiles of all students and of choosing the instructional approaches that will best support student learning. The approaches that teachers employ vary according to both the learning outcomes and the needs of the students, and teachers choose from and use a variety of accessible, equitable high-impact instructional practices.
The thoughtful use of these high-impact instructional practices – including knowing when to use them and how they might be combined to best support the achievement of specific math goals – is an essential component of effective math instruction. Researchers have found that the following practices consistently have a high impact on teaching and learning mathematics:
- Learning Goals, Success Criteria, and Descriptive Feedback. Learning goals and success criteria outline the intention for the lesson and how this intention will be achieved to ensure teachers and students have a clear and common understanding of what is being learned and what success looks like. The use of descriptive feedback involves providing students with the precise information they need in order to reach the intended learning goal. Using this practice makes all other practices more effective.
- Direct Instruction. This is a concise, intentional form of instruction that begins with a clear learning goal. It is not a lecture or a show-and-tell. Instead, direct instruction is a carefully planned and focused approach that uses questioning, activities, or brief demonstrations to guide learning, check for understanding, and make concepts clear. Direct instruction prioritizes feedback and formative assessment throughout the learning process and concludes with a clear summary of the learning.
- Problem-Solving Tasks and Experiences. It is an effective practice to use a problem, carefully chosen by the teacher or students, to introduce, clarify, or apply a concept or skill. This practice provides opportunities for students to demonstrate their agency by representing, connecting, and justifying their thinking. Students communicate and reason with one another and generate ideas that the teacher connects in order to highlight important concepts, refine existing understanding, eliminate unfruitful strategies, and advance learning.
- Teaching about Problem Solving. Teaching students about the process of problem solving makes explicit the critical thinking that problem solving requires. It involves teaching students to identify what is known and unknown and to draw on similarities between various types of problems. Teaching about problem solving involves using representations to model the problem-solving situation. This practice reinforces that problem solving requires perseverance and an awareness that mistakes can ultimately lead to growth in learning.
- Tools and Representations. The use of a variety of appropriate tools and representations supports a conceptual understanding of mathematics at all grade levels. Carefully chosen and used effectively, representations and tools such as manipulatives make math concepts accessible to a wide range of learners. At the same time, student interactions with representations and tools also give teachers insight into students’ thinking and learning.
- Math Conversations. Effective math conversations provide multiple opportunities for all students to engage in meaningful math talk by listening to and responding to the ideas of others. These conversations involve reasoning, proving, building on the thinking of others, defending and justifying their own thinking, and adjusting their perspectives as they build their mathematical understanding, confidence, and identity.
- Small-Group Instruction. A powerful strategy for moving student learning forward, small-group instruction involves targeted and timely mathematics instruction that meets the learning needs of specific students at appropriate times. By working with small and flexible groups, whether they are homogenous or heterogenous, the teachers can personalize learning in order to prevent gaps from developing, close gaps that already exist, or extend thinking. Small-group instruction also provides opportunities for teachers to learn more about student identities, experiences, and communities, which the teachers can use as a basis for their mathematics instruction.
- Deliberate Practice. Practice is best when it is purposeful and spaced over time. It must always follow understanding. This ensures that there is continual, consistent, and relevant feedback, so students know that they are practising correctly. Practice is an essential part of an effective mathematics program.
- Flexible Groupings. The intentional combination of large-group, small-group, partnered, and independent working arrangements, in response to student and class learning needs, can foster a rich mathematical learning environment. Creating flexible groupings in a mathematics class enables students to work independently of the teacher but with the support of their peers, and it strengthens collaboration and communication skills. Regardless of the size of the group, it is of utmost importance that individual students are accountable for and have ownership of their learning.
While a lesson may prominently feature one of these high-impact practices, other practices will inevitably also be involved. The practices are rarely used in isolation, nor is there any single “best” instructional practice. Teachers choose the right practice, for the right time, in order to create an optimal learning experience for all students. They use their knowledge of the students, a deep understanding of the curriculum and of the mathematics that underpins the expectations, and a variety of assessment strategies to determine which high-impact instructional practice, or combination of practices, best supports the students. These decisions are made continually throughout a lesson. The appropriate use of high-impact practices plays an important role in supporting student learning.
The mathematics curriculum was developed with the understanding that the strategic use of technology is part of a balanced mathematics program. Technology can extend and enrich teachers’ instructional strategies to support all students’ learning in mathematics. Technology, when used in a thoughtful manner, can support and foster the development of mathematical reasoning, problem solving, and communication.
When using technology to support the teaching and learning of mathematics, teachers consider the issues of student safety, privacy, ethical responsibility, equity and inclusion, and well-being.
The strategic use of technology to support the achievement of the curriculum expectations requires a strong understanding of:
- the mathematical concepts being addressed;
- high-impact teaching practices that can be used as appropriate to achieve the learning goals;
- the capacity of the chosen technology to augment the learning, and how to use this technology.
Technology can be used specifically to support the “doing” of mathematics (e.g., digital tools, computation devices, calculators, data-collection programs) or to facilitate access to information and allow better communication and collaboration (e.g., collaborative documents and web-based content that enable students to connect with experts and other students, near or far). Technology can support English language learners in accessing mathematics terminology and ways of solving problems in their first language. Assistive technologies are critical in enabling some students with special education needs to have equitable access to the curriculum and in supporting their learning, and must be provided in accordance with students’ Individual Education Plan (IEP), as required.
Teachers understand the importance of technology and how it can be leveraged to support learning and to ensure that the mathematics curriculum expectations can be met by all students. Additional information can be found in the “The Role of Information and Communications Technology” subsection of "Considerations for Program Planning”.
Classroom teachers are the key educators of students with special education needs. They have a responsibility to support all students in their learning, and they work collaboratively with special education teachers, where appropriate, to achieve this goal. Classroom teachers commit to assisting every student to prepare for living with the highest degree of independence possible. More information on planning for and assessing students with special education needs can be found in the “Planning for Students with Special Education Needs” subsection of “Considerations for Program Planning”.
Principles for Supporting Students with Special Education Needs
The following principles guide teachers in effectively planning and teaching mathematics programs to students with special education needs, and also benefit all students:
- The teacher plays a critical role in student success in mathematics.
- It is important for teachers to develop an understanding of the general principles of how students learn mathematics.
- The learning expectations outline developmentally appropriate key concepts and skills of mathematics across all of the strands that are interconnected and foundational.
- There is an important connection between procedural knowledge and conceptual understanding of mathematics.
- The use of concrete representations and tools is fundamental to learning mathematics in all grades and provides a way of representing both concepts and student understanding.
- The teaching and learning process involves ongoing assessment. Students with special education needs should be provided with various opportunities to demonstrate their learning and thinking in multiple ways.
An effective mathematics learning environment and program that addresses the mathematical learning needs of students with special education needs is purposefully planned with the principles of Universal Design for Learning in mind and integrates the following elements:
- knowing the student’s strengths, interests, motivations, and needs in mathematics learning in order to differentiate learning and make accommodations and modifications as outlined in the student’s Individual Education Plan;
- building the student’s confidence and a positive identity as a mathematics learner;
- valuing the student’s prior knowledge and connecting what the student knows with what the student needs to learn;
- focusing on the connections between broad concepts in mathematics;
- connecting mathematics with familiar, relevant, everyday situations and providing rich and meaningful learning contexts;
- fostering a positive attitude towards mathematics and an appreciation of mathematics through multimodal means, including through the use of assistive technology and the performance of authentic tasks;
- implementing research-informed instructional approaches (e.g., Concrete – Semi-Concrete – Representational – Abstract) when introducing new concepts to promote conceptual understanding, procedural accuracy, and fluency;
- creating a balance of explicit instruction, learning in flexible groupings, and independent learning. Each form of learning should take place in a safe, supportive, and stimulating environment while taking into consideration that some students may require more systematic and intensive support, more explicit and direct instruction before engaging in independent learning;
- providing environmental, assessment, and instructional accommodations that are specified in the student’s Individual Education Plan in order to maximize the student’s learning (e.g., making available learning tools such as manipulatives, resources, adapted game pieces, oversized tangrams, and calculators; ensuring access to assistive technology);
- building an inclusive community of learners and encouraging students with special education needs to participate in various mathematics-oriented class projects and activities;
- building partnerships with administrators and other teachers, particularly special education teachers, where available, to share expertise and knowledge of the curriculum expectations; co-develop content in the Individual Education Plan that is specific to mathematics; and systematically implement intervention strategies, as required, while making meaningful connections between school and home to ensure that what the student is learning in the school is relevant and can be practised and reinforced beyond the classroom.
English language learners are working to achieve the curriculum expectations in mathematics while they are acquiring English-language proficiency. An effective mathematics program that supports the success of English language learners is purposefully planned with the following considerations in mind.
- Students’ various linguistic identities are viewed as a critical resource in mathematics instruction and learning. This enables students to use their linguistic repertoire in a fluid and dynamic way, mixing and meshing languages to communicate. This translingual practice is creative and strategic, and allows students to communicate, interact, and connect with peers and teachers for a variety of purposes, such as when developing conceptual knowledge and when seeking clarity and understanding.
- Knowledge of English language learners’ mathematical strengths, interests, and identities, including their social and cultural backgrounds is important. These “funds of knowledge” are historically and culturally developed skills and assets that can be incorporated into mathematics learning to create a richer and more highly scaffolded learning experience for all students, promoting a positive, inclusive teaching and learning environment.
- In addition to assessing their level of English-language proficiency, an initial assessment of the math knowledge and skills of newcomer English language learners is required in Ontario schools.
- Differentiated instruction is essential in supporting English language learners, who face the dual challenge of learning new conceptual knowledge while acquiring English-language proficiency. Designing mathematics learning to have the right balance for English language learners is achieved through program adaptations (i.e., accommodations and/or modifications) that ensure the tasks are mathematically challenging, reflective of learning demands within the mathematics curriculum, and comprehensible and accessible to English language learners. Using the full range of a student’s language assets, including additional languages that a student speaks, reads, and writes, as a resource in the mathematics classroom supports access to their prior learning, reduces the language demands of the mathematics curriculum, and increases engagement;
- Working with students and their families and with available community supports allows for the multilingual representation of mathematics concepts to create relevant and real-life learning contexts and tasks.
In a supportive learning environment, scaffolding the learning of mathematics assessment and instruction offers English language learners the opportunity to:
- access their other language(s) (e.g., by using technology to access mathematical terminology and ways of solving problems in their first language), prior learning experiences, and background knowledge in mathematics;
- learn new mathematical concepts in authentic, meaningful, and familiar contexts;
- engage in open and parallel tasks to allow for multiple entry points for learning;
- work in a variety of settings that support co-learning and multiple opportunities to practice (e.g., with partners or in small groups, as part of cooperative learning, in group conferences);
- access the language of instruction during oral, written, and multimodal instruction and assessment, during questioning, and when encountering texts, learning tasks, and other activities in the mathematics program;
- use oral language in different strategically planned activities, such as “think-pair-share”, “turn-and-talk”, and “adding on”, to express their ideas and engage in mathematical discourse;
- develop both everyday and academic vocabulary, including specialized mathematics vocabulary in context, through rephrasing and recasting by the teacher and through using student-developed bilingual word banks or glossaries;
- practise using sentence frames adapted to their English-language proficiency levels to describe concepts, provide reasoning, hypothesize, make judgements, and explain their thinking;
- use a variety of concrete and/or digital learning tools to demonstrate their learning in mathematics in multiple ways (e.g., orally, visually, kinesthetically), through a range of representations (e.g., portfolios, displays, discussions, models), and in multiple languages (e.g., multilingual word walls and anchor charts);
- have their learning assessed in terms of the processes they use in multiple languages, both during the learning and through teachers’ observations and conversations.
Strategies used to differentiate instruction and assessment for English language learners in the mathematics classroom also benefit many other learners in the classroom, since programming is focused on leveraging all students’ strengths, meeting learners where they are in their learning, being aware of language demands in the mathematics program, and making learning visible. For example, different cultural approaches to solve mathematical problems can help students make connections to the Ontario curriculum and provide classmates with alternate ways of solving problems.
English language learners in English Literacy Development (ELD) programs in Grades 3 to 8 require accelerated support to develop both their literacy skills and their numeracy skills. These students have significant gaps in their education because of limited or interrupted prior schooling. They are learning key mathematical concepts missed in prior years in order to build a solid foundation of mathematics. At the same time, they are learning the academic language of mathematics in English while not having acquired developmentally appropriate literacy skills in their first language. Programming for these students is, therefore, highly differentiated and intensive. These students often require focused support over a longer period than students in English as a Second Language (ESL) programs. The use of the student’s oral competence in languages other than English is a non-negotiable scaffold. The strategies described above, such as the use of visuals, the development of everyday and academic vocabulary, the use of technology, and the use of oral competence, are essential in supporting student success in ELD programs and in mathematics.
Supporting English language learners is a shared responsibility. Collaboration with administrators and other teachers, particularly ESL/ELD teachers, where possible, contributes to creating equitable outcomes for English language learners. Additional information on planning for and assessing English language learners can be found in the “Planning for English Language Learners” subsection of "Considerations for Program Planning”.
Research indicates that there are groups of students who continue to experience systemic barriers to learning mathematics. Systemic barriers can result in inequitable outcomes, such as chronic underachievement and low confidence in mathematics. Achieving equitable outcomes in mathematics for all students requires educators to pay attention to these barriers and to how they can overlap and intersect, compounding their effect. Educators ensure that students have access to enrichment support, as necessary, and they capitalize on the rich cultural knowledge, experience, and competencies that all students bring to mathematics learning. When educators develop pedagogical practices that are differentiated, culturally relevant, and responsive, and hold high and appropriate expectations of students, they maximize the opportunity for all students to learn, and they create the conditions necessary to ensure that students have a positive identity as a mathematics learner and can succeed in mathematics and in all other subjects.
It is essential to develop practices that learn from and build on students’ cultural competencies and linguistic resources, recognizing that students bring a wealth of mathematical knowledge, information, experiences, and skills into the classroom, often in languages different from the language of instruction. Educators create the conditions for authentic mathematics experiences by connecting mathematics learning to students’ communities and lives; by respecting and harnessing students’ prior knowledge, experiences, strengths, and interests; and by acknowledging and actively reducing and eliminating the systemic barriers that some students face. Mathematics learning that is student-centered allows students to find relevance and meaning in what they are learning, to make real-life connections to the curriculum.
Mathematics classrooms also provide an opportunity for cross-curricular learning and for teaching about human rights. To create safe, inclusive, and engaging learning environments, educators must be committed to equity and inclusion for all students and to upholding and promoting human rights. Every student, regardless of their background, identity, or personal circumstances, has the right to have mathematics opportunities that allow them to succeed, personally and academically. In any mathematics classroom, it is crucial to acknowledge students’ multiple social identities and how students intersect with the world. Educators have an obligation to develop and nurture learning environments that are reflective of and responsive to students’ strengths, needs, cultures, and diverse lived experiences, and to set appropriate and high expectations for all.
Culturally Relevant and Responsive Pedagogy in Mathematics
Rich, high-quality instruction and tasks are the foundation of culturally relevant and responsive pedagogy (CRRP) in mathematics. In CRRP classrooms, teachers learn about their own identities and pay attention to how those identities affect their teaching, their ideas, and their biases. Teachers also learn about students’ identities, identifications, and/or affiliations and build on students’ ideas, questions, and interests to support the development of an engaging mathematics classroom community.
In mathematics spaces using CRRP, students are engaged in shaping much of the learning so that students have mathematical agency and feel invested in the outcomes. Students develop agency that motivates them to take ownership of their learning of, and progress in, mathematics. Teaching about diverse mathematical figures in history and from different global contexts enables students not only to see themselves reflected in mathematical learning – a key factor in developing students’ sense of self – but also to learn about others, and the multiple ways mathematics exists in all aspects of the world around them.
Culturally reflective and responsive teachers know that there is more than one way to develop a solution. Students are exposed to multiple ways of knowing and are encouraged to explore multiple ways of finding answers. For example, an Indigenous pedagogical approach emphasizes holistic, experiential learning; teacher modelling; and the use of collaborative and engaging activities. Teachers differentiate instruction and assessment opportunities to encourage different ways of learning, to allow all students to learn from and with each other, and to promote an awareness of and respect for the diverse and multiple ways of knowing that make up our classrooms, schools, and the world. When making connections between mathematics and real-life applications, teachers may work in partnership with Indigenous communities to co-teach. Teachers may respectfully incorporate Indigenous culturally specific examples as a way to meaningfully infuse Indigenous knowledge into the mathematics program. In this way, culturally specific examples can be used without cultural appropriation.
More information on equity and inclusive education can be found in the "Human Rights, Equity, and Inclusive Education" subsection of "Considerations for Program Planning".