# Mathematics

The aim of Saskatchewan’s K-12 mathematics program is to help students develop the understandings and abilities necessary to be confident and competent in thinking and working mathematically in their daily activities and ongoing learnings and work experiences. The mathematics program is intended to stimulate the spirit of inquiry within the context of mathematical thinking and reasoning. Defined below are four goals for K-12 Mathematics in Saskatchewan. The goals are broad statements that identify the characteristics of thinking and working mathematically. At every grade level, students’ learning should be building towards their attainment of these goals. Within each grade level, outcomes are directly related to the development of one or more of these goals. The instructional approaches used to promote student achievement of the grade level outcomes must, therefore, also promote student achievement with respect to the goals.

## Logical Thinking

Through their learning of K-12 Mathematics, students should develop and be able to apply mathematical reasoning processes, skills, and strategies to new situations and problems.

This goal encompasses processes and strategies that are foundational to understanding mathematics as a discipline. These processes and strategies include:

• observation
• inductive and deductive thinking
• proportional reasoning
• abstracting and generalizing
• exploring, identifying, and describing patterns
• verifying and proving
• exploring, identifying, and describing relationships
• modeling and representing (including concrete, oral, physical, pictorial, and symbolical representations)
• conjecturing and asking “what if” (mathematical play).

In order to develop logical thinking, students need to be actively involved in constructing their mathematical knowledge using the above strategies and processes. Inherent in each of these strategies and processes is student communication and the use of, and connections between, multiple representations.

## Number Sense

Through their learning of K-12 Mathematics, students should develop an understanding of the meaning of, relationships between, properties of, roles of, and representations (including symbolic) of numbers and apply this understanding to new situations and problems.

Foundational to students developing number sense is having ongoing experiences with:

• decomposing and composing of numbers
• relating different operations to each other
• modeling and representing numbers and operations (including concrete, oral, physical, pictorial, and symbolical representations)
• understanding the origins and need for different types of numbers
• recognizing operations on different number types as being the same operations
• understanding equality and inequality
• recognizing the variety of roles for numbers
• developing and understanding algebraic representations and manipulations as an extension of numbers
• looking for patterns and ways to describe those patterns numerically and algebraically.

Number sense goes well beyond being able to carry out calculations. In fact, in order for students to become flexible and confident in their calculation abilities, and to transfer those abilities to more abstract contexts, students must first develop a strong understanding of numbers in general. A deep understanding of the meaning, roles, comparison, and relationship between numbers is critical to the development of students’ number sense and their computational fluency.

## Spatial Sense

Through their learning of K-12 Mathematics, students should develop an understanding of 2-D shapes and 3-D objects, and the relationships between geometrical shapes and objects and numbers, and apply this understanding to new situations and problems.

Development of a strong spatial sense requires students to have ongoing experiences with:

• construction and deconstruction of 2-D shapes and 3-D objects
• investigations and generalizations about relationships between 2-D shapes and 3-D objects
• explorations and abstractions related to how numbers (and algebra) can be used to describe 2-D shapes and 3-D objects
• explorations and generalizations about the movement of 2-D shapes and 3-D objects
• explorations and generalizations regarding the dimensions of 2-D shapes and 3-D objects
• explorations, generalizations, and abstractions about different forms of measurement and their meaning.

Being able to communicate about 2-D shapes and 3-D objects is foundational to students’ geometrical and measurement understandings and abilities. Hands-on exploration of 3-D objects and the creation of conjectures based upon patterns that are discovered and tested should drive the students’ development of spatial sense, with formulas and definitions resulting from the students’ mathematical learnings.

## Mathematics as a Human Endeavour

Through their learning of K-12 Mathematics, students should develop an understanding of mathematics as a way of knowing the world that all humans are capable of with respect to their personal experiences and needs.

Developing an understanding of mathematics as a human endeavour requires students to engage in experiences that:

• value place-based knowledge and learning
• value learning from and with community
• encourage and value varying perspectives and approaches to mathematics
• recognize and value one’s evolving strengths and knowledge in learning and doing mathematics
• recognize and value the strengths and knowledge of others in doing mathematics
• value and honour reflection and sharing in the construction of mathematical understanding
• recognize errors as stepping stones towards further learning in mathematics
• require self-assessment and goal setting for mathematical learning
• support risk taking (mathematically and personally)
• build self-confidence related to mathematical insights and abilities
• encourage enjoyment, curiosity, and perseverance when encountering new problems
• create appreciation for the many layers, nuances, perspectives, and value of mathematics.

Students should be encouraged to challenge the boundaries of their experiences, and to view mathematics as a set of tools and ways of thinking that every society develops to meet their particular needs. This means that mathematics is a dynamic discipline in which logical thinking, number sense, and spatial sense form the backbone of all developments and those developments are determined by the contexts and needs of the time, place, and people.

The content found within the grade level outcomes for the K-12 Mathematics programs, and its applications, is first and foremost the vehicle through which students can achieve the four goals of K-12 Mathematics. Attainment of these four goals will result in students with the mathematical confidence and tools necessary to succeed in future mathematical endeavours.