What are the foundational principles in mathematics teaching and learning?

Math is about sense-making. "This the most fundamental idea that a teacher of mathematics needs to believe and act on. It is through the teacher's actions that every [learner] in his or her own way can come to believe this simple truth and, more importantly, believe that he or she is capable of making sense of mathematics. Helping students come to this belief should be the goal of every teacher. "

(Van de Walle, Teaching Student-Centred Mathematics, 2006, p. ix)

Considerations:

  • the First Peoples Principles of Learning and other ways of knowing contribute to a more holistic and experiential experience of mathematics and benefits all learners
  • students learn to think like mathematicians by being immersed in the "mathematical habits of mind":
    • persevering and using mathematics to solve problems in everyday life
    • recognizing 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

from BC Mathematics Curriculum

  • students learn to be mathematicians by embodying dispositions such as:
    • curiosity and a sense of wonder
    • playfulness
    • flexibility
    • sense making
    • resilience
  • problem-solving is foundational to the study of mathematics
  • an inquiry-based approach includes rich tasks and student problem posing, which nurtures engagement, curiosity and deep understanding
  • the language of mathematics supports learning and thinking like a mathematician
  • the foundational mathematical big ideas include:
    • number represents and describes quantity
    • development of computational fluency requires a strong sense of number
    • we use patterns to represent identified regularities and to form generalizations
    • we can describe, measure, and compare spatial relationships
    • analyzing data and chance enable to compare and interpret information
  • there is a progression of learning in mathematics
  • mathematics has underlying structures such as the associative property, the commutative property, and the distributive property

Resources:

The First Peoples Principles of Learning contribute to a more holistic and experiential experience of mathematics.

There are a number of resources to support the First Peoples Principles of Learning and other ways of knowing in the context of learning mathematics. Below is a list of online resources from a variety of organizations including the First Nations Education Steering Committee and projects such as Math Catcher, the Aboriginal Curriculum Integration Project through SD79, and Aboriginal Perspectives.

Students learn to think like mathematicians by being immersed in the mathematical habits of mind.

Assessing Early Numeracy.pdf

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 factor in the development of the well-educated citizen.

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

  • persevering and using mathematics to solve problems in everyday life
  • recognizing 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

From: BC's New Curriculum Introduction to Mathematics

Problem-solving is foundational to the study of mathematics.

George Polya, an influential mathematicican from the 1940s, described four steps in problem-solving in his book How to Solve It:

  • understand the problem
  • develop a plan and consider possible strategies
  • carry out the plan and use the strategies
  • look back and reflect

Mathematician Conrad Wolfram's approach to problem-solving involves four steps:

  • posing the right question
  • real world to math formulation
  • computation
  • math formulation back to the real-world

In this TED Talk Wolfram explains this approach as well as his arguments for teaching students mathematics through computer programming.

NCTMResearch_brief_14_-_Problem_Solving.pdf

Other resources on problem solving include this Research Brief put together by The National Council of Teachers of Mathematics on why it is important to student learning to teach with problem solving.

The British Columbia Provincial Graduation Numeracy Assessment (GNA)

Process for Solving Numeracy Tasks

This figure illustrates the numeracy processes involved in solving a numeracy task.

  • The task starts with a contextualized situation to provide context.
  • The contextualized situation is then interpreted to identify the real-world problem. From that, one or more mathematical approaches are applied (“mathematized”)
  • to create a relationship (or several) to solve the mathematical problem.
  • The mathematical problem is solved, and the resulting mathematical solution is analyzed and evaluated in context to determine if another cycle is needed.
  • Once the contextualized situation is resolved, a solution or recommendation is communicated.

An inquiry-based approach includes rich tasks and student problem posing, which nurtures engagement, curiosity and deep understanding

Jo Boaler - Enquiry Based Learning for Mathematics

Professor Jo Boaler works with secondary school students in the inQbate at the University of Sussex using Inquiry Based Learning methods for mathematics.

Creativity in Mathematics: Inquiry - Based Learning and the Moore Method

Explores the world of Inquiry-Based Learning and seeks to identify the reasons behind its celebrated success.

Enquiry-Based Learning in the Primary Classroom

Webinar

The language of mathematics supports learning and thinking like a mathematician.

"90% of math language happens only within the math classroom." - Krpan, 2018

Author Cathy Marks Krpan believes that through competency-based learning, students and teachers alike can deepen their mathematical understanding and share and impart that knowledge in and out of the math classroom. Teaching Math With Meaning takes a practical approach to embedding this deep learning in K to Grade 8 mathematics classrooms.

The Digest edition 2010-2 - Language in the mathematics classroom.pdf

The article Language in the Mathematics Classroom describes the role of language in Numeracy learning and the complexity of math language which often leads to ambiguity in conceptual understanding.

Basic Concept Words Related to Numeracy.docx

The document above lists many of the words we use related to numeracy.

There is a progression of learning in mathematics.

The following resources include professional resources, programs that provide a framework for teaching with progressions, learning continuums for mathematics and an online resource.

First Steps Mathematics is a series of resource books that will help teachers increase their capacity as mathematics educators. The program includes strategies to diagnose, plan, implement and judge the effectiveness of the learning experiences they provide for students. The link will take you to a site where you can access these resources for free.

PRIME stands for Professional Resources and Instruction for Mathematics Educators.

This resource provides courses and resources for teachers.

Learning Trajectories: A web-based tool for early childhood educators to learn about Learning Trajectories for math, review video clips of children’s thinking around the progression and activities to support development along the math trajectories. Educators must create an account in order to access this free resource.

ClementsSaramaLearning_trajectories_math.pdf

Number Worlds Learning Trajectories

Developmental Continuum Grade k-9.docx

Developmental Math Continuum Summary developed by the British Columbia Association of Math Teachers

Numeracy Continuum K-10 developed by the New South Wales Department of Education

Pearson Mathematics Learning Progression available in English & French

Math specialist Graham Fletcher has created a number of videos that demonstrate the progression of learning math on the following topics:

  • Early number and counting
  • Addition and subtraction
  • Progression of division
  • Progression of multiplication
  • Fractions: the meaning, equivalence, and comparison
  • NOTE: Graham is an American educator, so he is referencing the Common Core curriculum (ie. don't pay attention to the grades!)