Mathematics

Introduction to Mathematics

Mathematics is integral to every aspect of daily life. Mathematical skills are essential for solving problems in most areas of life and are part of human history. All peoples have used and continue to use mathematical knowledge and competencies to make sense of the world around them.

Mathematical values and habits of mind go beyond numbers and symbols; they help us connect, create, communicate, visualize, and reason, as part of the complex process of problem solving. These habits of mind are valuable when analyzing both novel and complex problems from a variety of perspectives, considering possible solutions, and evaluating the effectiveness of the solutions. When developed early in life, mathematical habits of mind help us see the math in the world around us and help to generate confidence in our ability to solve everyday problems without doubt or fear of math.

Observing, learning, and engaging in mathematical thinking empowers us to make sense of our world. For example, exploring the logic of mathematics through puzzles and games can foster a constructive mathematical disposition and result in a self-motivated and confident student with unique and individualized mathematical perspectives. Whether students choose to pursue a deeper or broader study in mathematics, the design of the Mathematics curriculum ensures that they are able to pursue their individual interests and passions while establishing a strong mathematical foundation.

Flexible teaching and learning

The Mathematics curriculum allows for instructional flexibility. For example, the curriculum components may be combined to provide a diverse range of learning opportunities. Within and across grades, there are multiple ways to combine learning standards to create lessons, units, and learning experiences, encouraging any and all approaches that support the growth and development of students’ mathematical understandings and skills.

The focus on flexible teaching and learning enables teachers to confidently choose the strategies, resources, and applications best suited to the needs of students in their local setting (e.g., embedding mathematics in issues, projects, and passions relevant to the local community). It enables teachers to focus on “hands-on” experiential learning, by incorporating  the learning of foundational skills through opportunities to encounter math in a wide variety of situational contexts.

Explicit financial literacy components are included throughout the K-12 curriculum, as part of building a strong foundation of mathematical understanding and skills for every student. Regardless of the pathway students choose in Grades 11 and 12, they will share a common experience in the Mathematics curriculum that includes mathematical reasoning and probability/statistics, along with financial literacy components that have been customized to fit each area of specialty:

  • Grade 11 courses (with the exception of History of Mathematics) share similar financial literacy concepts, with the structure and emphasis differing, based on the course.
  • In Grade 12, both Apprenticeship Mathematics 12 and Foundations of Mathematics 12 continue financial literacy education. Apprenticeship Mathematics 12 emphasizes financial learning relevant to those pursuing the post-secondary apprenticeship path, and Foundations of Mathematics 12 continues to broaden student understanding in personal financial decision making.

Features of the Mathematics curriculum

The Mathematics curriculum is designed to build on students’ mathematics knowledge and enables them to apply this knowledge to a broad range of situations encountered in everyday life. This is facilitated by condensing the learning standards, focusing on flexible teaching and learning within relevant situational contexts, and continuing to develop a strong foundation of mathematical understandings and skills, as just one part of an interdisciplinary set of problem-solving, exploratory, and investigative skills and knowledge.

Design of the Mathematics curriculum

The Mathematics curriculum has the same format as all other areas of learning. Three curricular elements – the Big Ideas, Curricular Competencies, and Content – link the knowing, doing, and understanding of mathematics learning. Elaborations support each curricular area by providing suggestions, definitions, and clarifications to better support teaching and learning. More information on the curriculum model is available at https://www.curriculum.gov.bc.ca/curriculum/overview.

In all courses, the focus is on real-life, relevant contexts for learning, and all courses take a problem-solving approach. The Curricular Competencies for all Mathematics courses are organized around a problem-solving process adapted from the National Council of Teachers of Mathematics (NCTM), which supports inquiry, the development of thinking strategies, and the explanation and justification of mathematical ideas.

Big Ideas

The Big Ideas are generalizations and principles discovered through experiencing the Content and Curricular Competencies of the curriculum – the “Understand” component of the Know-Do-Understand model of learning. Big Ideas represent the “aha!” and the “so what?” of the curriculum – the deeper learning.

The Big Ideas of the Mathematics curriculum highlight the progression of related skills and concepts. For each area of mathematics in Kindergarten through Grade 9 (K-9) – computational fluency, number, patterns and relations, spatial sense, and statistics and probability – important concepts are introduced in Kindergarten, growing with students and expanding in scope and depth of learning as they progress through the grades. For Grades 10 through 12, students have the opportunity to further explore their passions and interests through diverse Mathematics courses. In each of these, specialized learning builds on the K-9 progression of skills and concepts.

The chart below shows an example of the progression of number from Kindergarten through Grade 9.

Big Ideas

Number: Number represents and describes quantity

K

Numbers represent quantities that can be decomposed into smaller parts.

1

Numbers to 20 represent quantities that can be decomposed into 10s and 1s.

2

Numbers to 100 represent quantities that can be decomposed into 10s and 1s.

3

Fractions are a type of number that can represent quantities.

4

Fractions and decimals are types of numbers that can represent quantities.

5

Numbers describe quantities that can be represented by equivalent fractions.

6

Mixed numbers and decimal numbers represent quantities that can be decomposed into parts and wholes.

7

Decimals, fractions, and percents are used to represent and describe parts and wholes of numbers.

8

Number represents, describes, and compares the quantities of ratios, rates, and percents.

9

The principles and processes underlying operations with numbers apply equally to algebraic situations and can be described and analyzed.

 

Curricular Competencies

The Core Competencies – Thinking, Communication, and Personal and Social – are embedded in the Curricular Competencies. The Curricular Competencies introduced in Kindergarten have been expanded based on a developmental continuum throughout the grades that is focused on what students can do with their Content knowledge in mathematics. Students also build on their Curricular Competencies from year to year. The example below shows how the Curricular Competencies grow with students and expand the scope and depth of learning.


 

Curricular Comp.

K

3

6

9

12
Pre-calculus

 

Use reasoning to explore and make connections

Use reasoning to explore and make connections

Use reasoning and logic to explore, analyze, and apply mathematical ideas

Use reasoning and logic to explore, analyze, and apply mathematical ideas

Explore, analyze, and apply mathematical ideas using reason, technology, and other tools

 

Content

The Content is concept-based and reflects what students should know. It identifies the concepts or topics that students will learn about at each grade level. The Content acts as both a supporting structure intended to assist students in demonstrating the Curricular Competencies and a foundational element leading students to the Big Ideas. Examples of Content learning standards are shown below.

 

Content

K

3

6

9

12
Pre-calculus

 

number concepts to 10

number concepts to 1000

small to large numbers (thousandths to billions)

operations with rational numbers (addition, subtraction, multiplication, division, and order of operations)

polynomial functions and equations

 

Elaborations

There are elaborations (included as hyperlinks) for many of the learning standards in the Mathematics curriculum. The elaborations take the form of explanations, definitions, and clarifications. They provide additional information and support for both teachers and students and can serve as potential places to begin teaching and learning. Examples of elaborations are shown below.

 

K

3

6

9

12
Pre-calculus

Content

equality as a balance and inequality as an imbalance

addition and subtraction to 1000

multiplication and division facts to 100 (developing computational fluency)

operations with rational numbers (addition, subtraction, multiplication, division, and order of operations)

polynomial functions and equations

Elaboration

equality as a balance:

  • modelling equality as balanced and inequality as imbalanced using concrete and visual models (e.g., using a pan balance with cubes on each side to show equal and not equal) fish drying and sharing

addition and subtraction:

  • using flexible computation strategies, involving taking apart (e.g., decomposing using friendly numbers and compensating) and combining numbers in a variety of ways, regrouping
  • estimating sums and differences of all operations to 1000
  • using addition and subtraction in real-life contexts and problem-based situations
  • whole-class number talks

facts to 100:

  • mental math strategies (e.g., the double-double strategy to multiply 23 x 4)

operations:
includes brackets and exponents
simplifying (-3/4) ÷ 1/5 + ((-1/3) x (-5/2))
simplifying 1 – 2 x (4/5)2
paddle making

polynomial: 

  • factoring, including the factor theorem and the remainder theorem  
  • graphing and the characteristics of a graph (e.g., degree, extrema, zeros, end-behaviour)
  • solving equations algebraically and graphically

 

Important considerations

Inquiry in mathematics

The Mathematics curriculum continues to support the application of foundational math skills in problem solving. It is important for students to be able to approach problem solving with confidence. A problem-solving model provides students with the necessary skills to read a problem, choose from a variety of appropriate strategies, apply a strategy to solve the problem, and then reflect on the efficiency and accuracy of the strategy to explain the answer.

Mathematical habits of mind

Extensive research indicates that for students to develop mathematical habits of mind they must encounter and interact in intentional learning settings. Classroom design combined with active participation strategies will enhance student learning, increase achievement, and contribute to the development of well-educated citizens.

Students who have developed mathematical habits of mind exhibit expertise in:

  • persevering and using mathematics to solve problems in everyday life
  • recognizing that there are multiple ways to solve a problem
  • demonstrating respect for diversity in approaches to solving problems
  • choosing and using appropriate strategies and tools
  • pursuing accuracy in problem solving

First Peoples knowledge and perspectives

The Ministry of Education is dedicated to ensuring that the cultures and contributions of First Peoples in British Columbia are reflected in all provincial curricula.

First Peoples Principles of Learning

The First Peoples Principles of Learning have been affirmed within First Peoples societies to guide the teaching and learning of provincial curricula. Because these principles of learning represent an attempt to identify common elements in the varied teaching and learning approaches that prevail within particular First Peoples societies, it must be recognized that they do not capture the full reality of the approach used in any single First Peoples society.

The First Peoples Principles of Learning greatly influence the B.C. curriculum and are woven throughout. They lend themselves well to mathematical learning, as they promote experiential and reflexive learning, as well as self-advocacy and personal responsibility in students. They help create classroom experiences based on the concepts of community, shared learning, and trust, all of which are vital to learning.

To address First Peoples content and perspectives in the classroom in a way that is accurate and that respectfully reflects First Peoples concepts of teaching and learning, teachers are strongly encouraged to seek the advice and support of members of local First Peoples communities. As First Peoples communities are diverse in terms of language, culture, and available resources, each community will have its own unique protocol to gain support for integration of local knowledge and expertise. Permission for the use or translation of cultural materials or practices should be obtained through consultation with individuals, families, and other community members. This authorization should be obtained prior to the use of any educational plans or materials.