“The environment…conveys the message that this is a place where adults have thought about the quality and instructive power of space. The layout of the physical space is welcoming and fosters encounters, communication, and relationships. The arrangement of structures, objects, and activities encourages choices, problem solving, and discoveries in the process of learning.” 1
Mathematical learning environments include effective interplay of:2
- Reflection and metacognition
- Exploration of patterns and relationships
- Sharing of ideas and problems
- Consideration of different perspectives
- Decision making
- Generalization and abstraction
- Verifying and proving
- Modelling and representing
- Making connections
Just as it is important to move our pedagogical focus from a focus on the teaching to a focus on the learning, we also wish to focus on creating learners rather than students of mathematics.
Researchers have established a clear connection between the attitudes that students have towards mathematics and how well they perform mathematically. For example, Pajares and Kranzler (1995) found that the belief that one is capable of doing mathematics (self-efficacy) is a better predictor of senior mathematics achievement than is prior math achievement. Similarly, Bruce and Ross (2010) established that Ontario students’ belief in their mathematical abilities is an important determinant of their achievement.3
Growth Mindset in Mathematics:4
In her study of four high-performing Grade 9 Applied Mathematics classrooms, Macaulay (2015) found that many students began the course with a fractured relationship with mathematics and mathematics learning and a poor sense of themselves as math learners. Their teachers made it a priority to build student confidence and saw their first order of business as helping students see themselves as capable and competent people and as capable and competent math learners. Once students began to think of themselves in this way, they began to thrive in the classroom.5
Approaching math with a holistic view of the student and the intent to encourage students to learn from mistakes, to persevere in overcoming challenges and to recognize and enjoy success, will invite students of mathematics to reach their potential as they see themselves as capable learners.
For a deeper dive into a conversation around Mathematical Mindsets, see: Mathematical Mindsets: A Conversation with Glen Aikenhead.
When Indigenous learners sense teachers not insisting that they meet the teacher’s high expectations, these learners quickly self-label as being less worthy and not respected. B.C.’s provincial Auditor-General reported, “Our 2015 report highlighted the impact of the racism of low expectations [for Indigenous learners]” (Bellringer, 2019, p. 13, original emphasis).6
Questions for Reflection
How does the classroom environment reflect students as capable, competent Western Math learners with respect to their personal experiences and needs?
How do my students see themselves and their interests reflected in the classroom environment?
In what ways do I share with families to help them support their child’s mathematical and learning development?
How do I structure my class to support collaboration between students?
Positive Nurturing Relationships:
There are many aspects and reasons to foster positive relationships in a Math classroom. It is important to create positive, nurturing relationships between student and teacher. It is essential to develop positive relationships between student and subject area, commonly referred to as a growth mindset. These relationships are vital to student success.
Gail Boushey and Joan Moser (Math Daily Three)6 suggest a framework to facilitate independence in Math. They suggest that one of the most important framework components is to build relationships whereby teachers believe all students are capable and worthy of learning math through trust and respect. The framework allows teachers to provide opportunities for all students to develop their skills as independent learners.
John Hattie’s research shows the importance of the relationship between teacher and student. One of the most effective indicators with an effect size of 0.72 is the teacher-student relationship.7
When there is a positive teacher-student relationship, students feel safe and there is a strong bond of trust within the classroom. Students are not afraid to take risks and understand that making errors are all part of the learning process. Students are more likely to feel positive about school and have a greater chance of developing a true love for learning.
Jo Boaler8 suggests the following “norms” for a successful mathematics classroom:
Oftentimes students perceive messages that they are not good at Math or they hear from other people who say they are not good at Math. Research does not support the idea that only some people are born with a highly developed math brain. There is a spectrum of flexible predispositions that people hold with respect to further developing, for example, their math proficiency, their mathematical self-identify, their mathematical curiosity, etc. (See also Supporting Families).
Peter Liljedahl’s9 research and work with thinking classrooms promotes active student participation and engagement while working collaboratively on well-designed tasks that include a productive struggle. The teacher’s role is to help create a positive environment where students are encouraged to collaborate and creativity is celebrated and required.
Interactions between students and teachers can impact the success a student can achieve. Through meaningful feedback, growth mindset messages, positive relationships with students can determine and impact the success a child can experience.
When these relationships include an opportunity to fail, an opportunity to struggle, an opportunity to contemplate, an opportunity to celebrate success, a student can develop important skills in Math including problem solving, collaboration, creativity, and persistence, among other important traits and behaviours.
Fostering a Math-Rich Classroom
“The learning climate must include positive personal relationships that enhance development through meaningful conversations, a sense of care for the whole student that goes beyond academic concerns. The nurturing classroom meets the holistic needs of students — social, emotional, physical, intellectual, and spiritual” (Williams, n.d., p. 24).
Four of the important characteristics of a math classroom are:
- Establishing a Mathematical Mindset;
- Constructing Understanding;
- Supporting Productive Struggle; and
- Developing Classroom discourse.
Establishing a Mathematical Mindset
Much needs to be done about our attitude toward mathematics. It seems it is socially acceptable and sometimes even desirable to express an ineptitude toward math. We need to adopt a mathematical mindset. Mathematical literacy is for everyone!
When Indigenous learners sense teachers not insisting that they meet the teacher’s high expectations, these learners quickly self-label as being less worthy and not respected. B.C.’s provincial Auditor-General reported, “Our 2015 report highlighted the impact of the racism of low expectations [for Indigenous learners]” (Bellringer, 2019, p. 13, original emphasis).6
Questions for Reflection
Do I believe there is a spectrum of flexible predispositions which people hold with respect to further developing their math proficiency, their math self-identity, their math curiosity, etc.?
Do I reinforce respectful listening and sharing in a safe environment?
Do I have students set goals for themselves and self-monitor their progress?
Do I avoid competition and timed tasks because I recognize that speed does not equate to fluency?
Developing a Growth Mindset
| Instead Of… | Try Thinking… |
|---|---|
| I’m not good at this. | What am I missing? |
| I give up. | I’ll use a different strategy. |
| It’s good enough. | Is this really my best work? |
| I can’t make this any better. | I can always improve. |
| This is too hard. | This may take a little time. |
| I made a mistake. | Mistakes help me learn. |
| I just can’t do this. | I can train my brain. |
| I’ll never be that smart. | I can learn how to do this. |
| My plan doesn’t work. | There’s always plan B. |
| My friend can do it. | I will learn from them. |
Constructing understanding
In a constructivist classroom, students build their own understanding of mathematical concepts through their experiences with math problems. Students use mathematical models as needed to help explain, evaluate, and communicate the concept (“Constructivism in the Classroom,” n.d.)
Do I have students share their understanding of a concept before I share mine?
Do I use my students’ responses to determine future lessons, strategies and content presented?
Do I plan activities that allow students to experience contradictions to their current understandings and then discuss their new learning?
Do I allow wait time when asking questions?
Do I encourage students to show initiative and autonomy in their work?
Developing Classroom Discourse
Classroom discourse is an essential aspect of a math classroom. Students are grouped to ensure academic diversity. They then work with their peers to solve complex and rich tasks. The discussions emerging from this situation are essential to the students’ learning.10
In a constructivist classroom, students build their own understanding of mathematical concepts through their experiences with math problems. Students use mathematical models as needed to help explain, evaluate, and communicate the concept (“Constructivism in the Classroom,” n.d.)
Do I ask open-ended questions?
Do I encourage students to ask questions of each other?
Do I expect students to explain all their answers, regardless of whether the answer is correct?
Do I model listening attentively to all answers?
Do I use group or whole class discussions to determine whether an answer is correct (rather than being the authority)?
Do I model respect for multiple strategies for solving a problem?
Math Tools & Resources
Manipulatives:
Math manipulatives are physical objects students can use to better understand abstract mathematical concepts. Students from all grade levels can benefit from the use of physical and virtual manipulatives in math instruction. These interactive tools offer engaging ways for students to actively construct their own mathematical learning and communicate this learning to their peers and teachers.
The opportunity for students to conceptualize abstract math concepts by seeing concrete representations can make all the difference in the depth of their understanding.
Tips for Using Manipulatives
- Careful consideration must be given to the type of manipulative and how it will be used in order for it to be a valuable tool which will enhance the learning of the math concept.
- A variety of math manipulatives or tools should be available to support the diverse learning and understanding of all students.
- Expectations for using the math manipulatives and tools, including how to retrieve and return materials, need to be clearly communicated. Manipulatives are not toys but are powerful learning tools which build conceptual understanding of mathematics.
- Store math manipulatives in an easily-accessible location which are available for students to use as needed.
There are many kinds of manipulatives that are commercially available in addition to manipulatives which can be made from common objects that are purchased inexpensively or found in one’s environment. Experiment and find manipulatives that work well for your students and are best suited for the mathematical concept.
Examples of Manipulatives:
| Centimeter Cubes | Base 10 Blocks | Measuring Tapes |
| Dominoes | Colour Tiles | Anglegs |
| Dice | Pattern Blocks | Attribute Blocks |
| Polydrons Dice (multi-sided dice) | Tangrams | Mini Whiteboards and Markers |
| Number Spinners | Connecting/Unifix Cubes | Reflectors |
| Two-colour counters | Fraction Strips & Towers | Balances |
| Algebra Tiles | Fraction Circles | Playing Cards |
| Relational/Cuisenaire Rods. | Rekenreks | Hundred Charts |
| Frames | Calculators | Number Lines |
| Geoboards | Geometric Solids | Counters, such as beans |
| Craft Sticks | Straws | Paper Plates |
Suggestions on Teaching with Manipulatives
Concepts to teach with counters:
| Counting | Number recognition | Number correspondence |
| Ordinal numbers | More, less and equivalence | The concept of ten (use counters with ten frames) |
| Addition and subtraction | Multiplication and division |
Concepts to teach with Base 10 Blocks
| Place Value | Regrouping | Addition |
| Subtraction | Multiplication | Division |
| Division with remainders | Zero as place-holder | Fractions |
| Decimals | Expanded notation |
Concepts to teach with Pattern Blocks
| Patterns | Shapes | Geometric designs |
| Spatial Relations | Types of angles | Area, parallel and perpendicular lines, |
| other geometry concept |
Concepts to teach with a Bucket Balance
| Explore volume | Compare solids and liquids | Measure mass |
| Measuring | Estimating | Cause and effect |
| Explore volume vs. mass |
Concepts to teach with Fraction Tiles and Circles
| the meaning of numerators and denominators | how fractions work | fraction addition and subtraction |
| equivalent fractions | multiplication of fractions | division of fractions |
Concepts to teach with Ten Frames
| Counting | Subitization | Number correspondence |
| More, less and equivalence | Addition and subtraction | Components of ten |
| Regrouping | Decomposition of large numbers | Problem solving |
Concepts to teach with Linking Cubes
| Patterns | Number correspondence | Counting |
| Skip counting | Comparisons | Measurement |
| Regrouping | Sorting | Addition and subtraction |
| Multiplication and division | Squared and cubed numbers | Graphing |
Concepts to teach with Geoboards
| Plane Shapes (squares, rectangles, triangles, parallelograms, trapezoids, etc…) | Symmetry | Rotation |
| Translation | Reflection | Coordinate Planes |
| Graphing | Types of angles | Problem solving |
| Spatial relations |
Literature
Literature is an ideal way to help your students see the importance of math in their daily lives. Using age-appropriate, quality literature can allow for students to conceptualize their mathematical thinking and enhance their learning of math concepts. When selecting literature into math instruction, consider these three questions:
- Is the book of high quality from a literary perspective?
- Does the book present content that is mathematically sound and grade-level appropriate?
- Is the book effective for helping students learn to think and reason mathematically?
Tips for Using Literature in Math Instruction11
- Allow time for students to enjoy the book. Savour the text and examine the illustrations.
- Provide the opportunity for class discussion.
- Now, during a re-read of the book, you can shift the students’ attention to make the math connection.
- After the lesson, make the book available for students to revisit on their own.
Picture Books for Math Instruction (suggested examples, not recommendations)
Literature in High School Math (Click for suggested examples of possible texts):
Many picture books, while not exclusively so, lend themselves well to a Middle or Elementary school context. There is an ever growing trend in the literature to take a more holistic approach to interdisciplinary literacy. While novel “studies” may seem overwhelming in terms of time, using literature to create a real life context for math is less ominous. See this example of ways that one math teacher has used current literature (and movies) to bring real life application to math concepts.
Recurrent Learning Strengths
Contrary to popular belief, there is little evidence that a stereotype “learning style” for Indigenous learners exists. Instead, evidence points to their “recurrent learning strengths” that tend to be found among Indigenous learners. These strengths include:
- holistic more than analytic
- visual more than verbal
- oral more than written
- practical more than theoretical
- reflective more than trial-and-error
- contextual more than non-contextual
- personally relational more than an impersonal acquisition of isolated facts and algorithms
- experiential more than passive
- oriented to storytelling more than didactic sessions, and
- taking time to reflect more than quickly coming to an answer.
These recurrent learning strengths are evident in non-Indigenous learners to varying degrees, as well (Aikenhead et al., 2014, p. 135).13
Some Considerations for Fostering a Supportive Math Learning Environment:14
- Create a community of mathematics learners that includes the educator.
- Insist that we are all “math people” and seek examples to show practical examples that illustrate this.
- Value the thinking that all students bring to the classroom.
- Help students to appreciate that errors and failed attempts are opportunities for learning and have value.
- Focus on understanding so that students recognize that mathematics must always make sense to them.
- See the student as a whole person, paying attention to all developmental domains when planning instruction, assessment and learning (e.g., provide opportunities to move while learning, plan for supportive social interactions, consider the emotional impact of instruction).
- Make learning the goal by supporting every student in playing an active role in his/her learning.
- Be careful about offering unsolicited help, and especially only targeting low achievers for assistance. Listen to each student about his/her goals and needs.
- Provide cognitively challenging tasks and take the students’ strengths, needs, interests and views into account when planning learning opportunities.
- Provide timely and descriptive feedback that will help students to improve.
- Inspire students to see math in the world around them.
“Reconciliation is not an event. It’s something that needs to enter into the way we do things.” (John Ralston Saul, 2014)15
| Link | Description |
|---|---|
https://www.facebook.com/ThinkIndigenousOnlineEd/ | Think Indigenous — Online Indigenous Education K-816 Lamarr Oksasikewiyin, Marty Scott, Kevin Lewis |
https://www.facebook.com/NCCIECanada/videos/2872719279480561/ | The National Centre for Collaboration in Indigenous Education17 Approaches to Maths & Sciences in Indigenous Learning |
| http://empoweringthespirit.ca/pedagogy/ | PEDAGOGY that embraces Indigenous ways of knowing are fostered by approaches to teaching and learning that include purposeful thinking about people, places and processes. The word Etuaptmumk, or Two-Eyed Seeing, communicates the belief that the most beneficial outcome occurs when we consider multiple perspectives in understanding and exploring ideas. Two-Eyed Seeing helps us to acknowledge the idea of wholeness, a part of many Indigenous knowledge systems: seeing things through Indigenous perspectives (represented as one whole eye), while also seeing western ways of knowing (also represented as a whole eye), inviting these two eyes to work together as they do in binocular vision. A weaving back and forth between knowledge systems that embrace a flow between the strengths of the two ways, to best suit the circumstances, strengthens the approach further. |
https://guides.library.utoronto.ca/indigenouseducation/lessonplans | Math lesson plans with a focus on Social Justice Education. |
| http://go.utlib.ca/cat/6297294 | Infusing Indigenous Perspectives in K-12 Teaching, University of Toronto; “Math that Matters”. |
| https://www.youtube.com/watch?v=p1rlphwI6RM&feature=youtu.be. | An 18-minute conversation between consultant Sharon Meyer and teacher Serena Palmer about her developing and teaching math lessons that incorporate Indigenous mathematizing. The teacher’s classroom environment becomes apparent during the discussion. |
1Curtis, & Carter. (2020). Retrieved 5 July 2020, from https://pubsaskdev.blob.core.windows.net/pubsask-prod/86149/86149-86149-Creating_Early_Learning_Environments_(003).pdf (p.13)
2SK Ministry of Education, 2010, p. 23
3Capacity Building K–12. (2020). Retrieved 5 July 2020, from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/math_classroom.html
4Ohio State University. (2014, December 10). It doesn’t add up: People who say they are good at math, but aren’t. ScienceDaily. Retrieved June 19, 2020 from www.sciencedaily.com/releases/2014/12/141210074142.htm
5Macaulay, A. (2015). Effective practices in Grade 9 Applied Mathematics (Doctoral dissertation).
6Bellringer, C. (2019). Progress audit: The education of Aboriginal students in the B.C. public school system. Victoria, B.C.: Office of the Auditor General of British Columbia.
6Boushey, G., & Moser, J. (2020). 1. Framework | TheDailyCAFE.com. Retrieved 4 July 2020, from https://www.thedailycafe.com/math-daily-3/math-essential-elements/understand/1-framework
7John Hattie (2009) Hattie, Visible Learning, p. 118
8Boaler, J. (2020). Retrieved 4 July 2020, from http://www.youcubed.org/wp-content/uploads/Positive-Classroom-Norms2.pdf
9Liljedahl, P. (2020). Peter Liljedahl. Retrieved 5 July 2020, from http://www.peterliljedahl.com/
10 (Hattie et al., 2017).
11(Information from Daily 3 Math and Marilyn Burns)
12Using Math Manipulatives to Build Understanding. (2020). Retrieved 5 July 2020, from http://mightymath.weebly.com/uploads/5/8/5/7/5857657/using_math_ manipulatives_full_file_print_with_markups1.pdf
13Aikenhead, G., Brokofsky, J., Bodnar, T., Clark, C., Foley, C., … Strange, G. (2014). Enhancing school science with Indigenous knowledge: What we know from teachers and research. Saskatoon, Canada: Saskatoon Public School Division with Amazon.ca. Retrieved from http://www.amazon.ca/Enhancing-School-Science-Indigenous-Knowledge/dp/149957343X.
14Edu.gov.on.ca. (2020). Retrieved from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/math-classroom2018.pdf
15Saul, J. The comeback (p. 260). Canada: Viking.
16Think Indigenous – Online Indigenous Education K-8. Facebook.com. (2020). Retrieved 22 May 2020, from https://www.facebook.com/ThinkIndigenousOnlineEd/.
17NCCIE – Approaches to Maths & Sciences in Indigenous Learning. Facebook Watch. (2020). Retrieved 22 May 2020, from https://www.facebook.com/NCCIECanada/videos/2872719279480561/.
