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A Pedagogical System

A Framework of Effective Mathematics Instruction

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​​​​There is an ever growing body of research that mathematics educators can leverage as they plan and refine their pedagogical approach in an attempt to design an effective mathematics program.​ Through a meta-analysis by Anthony and Walshaw (2007), they categorized their findings into Characteristics of Effective Teaching of Mathematics as represented in this graphic:

Principles of effective pedagogy of mathematics anthony walshaw 2007.png


Effective mathematical pedagogy is a coherent system rather than a set of discrete, interchangeable strategies. The way instructional tasks are realized in the classroom and experienced by students depends on the classroom atmosphere, the tools and representations available for them to use and the nature and focus of classroom discourse."

Anthony and Walshaw, 2007


The 4 T's

From the ten characteristics outlined in the graphic, a team of Student Achievement Officers at the Ministry of Education found that these characteristics could be grouped into four broad categories known as "The 4 T's" listed below:

  • TRUST: Non-Threatening Classroom Environment
  • TASKS: Worthwhile Tasks
  • TOOLS: Tools and Representations
  • TALK: Mathematical Discourse

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Unpacking The 4 T's

Trust: A Non-Threatening Classroom Enivronment


“Teachers who truly care about their students have high yet realistic expectations about enhancing students’ capacity to think, reason, communicate and reflect upon their own and others’ understanding.”
 ~ Anthony and Walshaw

A non-threatening classroom atmosphere is one in which there is an acknowledgement that all students have the capacity to become powerful mathematics learners within a culture of trust and academic press. In this caring classroom community, educators engage in interrelationships that help create spaces for learners to develop their mathematical identities and proficiencies.


Tasks: Worthwhile Instructional Tasks


“Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.”
 ~ Principles to Actions (2014), pg.17

The most productive, worthwhile tasks and activities:
  • focus on the solution of genuine mathematical problems, through the use of real or imaginary contexts, which illustrate the way the mathematics is used; 
  • allow students to access important mathematical concepts and relationships; 
  • encourage students to use varied approaches and strategies to make sense of and solve tasks; 
  • provide students with opportunities to engage in productive struggle as they grapple with mathematical ideas and relationships.


Tools: Tools and Representations


“Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.”
 ~ Principles to Actions (2014), pg.24

Tools and representations are used as thinking spaces for students to help organize mathematical thinking and to communicate mathematical ideas. When students are able to make connections between a range of representations through carefully sequenced examples, they will deepen their understanding of key mathematical concepts. Symbolic representations characteristic of mathematics include the number system, algebraic symbolism, graphs, diagrams, models, equations, notations for fractions, pictorial images, analogies, metaphors, models (such as pizzas, chocolate bars and ten frames), examples, stories, illustrations, textbooks, rulers, clocks, calendars, and technology, such as computers, calculators, digital, and problem contexts.


Talk: Classroom Discourse


“Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments.”
 ~ Principles to Actions (2014), pg.29

Anthony & Walshaw describe classroom discourse as more than developing a respectful, trusting and non-threatening climate for discussion and problem solving. Effective classroom discourse includes reasoning, argumentation, and justification of mathematical concepts. Discourse includes the purposeful exchange of ideas through classroom discussion, in order to share ideas, clarify understandings, construct convincing arguments, develop mathematical language, articulate comprehensible explanations to justify mathematical concepts, and to learn to see other mathematical perspectives. Mathematical discourse can be communicated through a variety of forms, for example, verbal, visual, and written. Teachers engage students in classroom discourse by valuing students’ ideas, exploring students’ answers, incorporating students’ background knowledge, and encouraging explicit student communication about mathematical learnings.

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