Four important characteristics in fostering a mathematically rich classroom are:

- Establishing a Mathematical Mindset
- Constructing Understanding
- Supporting Productive Struggle
- Developing Classroom Discourse

- Mathematical learning environments include effective interplay of:
- Reflection and metacognition.
- Exploration of patterns and relationships.
- Sharing of ideas and problems.
- Consideration of different perspectives.
- Decision making.
- Generalization and abstraction.
- Veryifying and proving.
- Modeling and representing.
- Making connections.

## 1. 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!

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?

## 2. 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^{1}

“Mathematics is learned when students are engaged in strategic play with mathematical concepts and differing perspectives.”SK Ministry of Ed, Math 3 (2009), p. 23

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.

Questions for reflection

- 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?

## 3. Supporting Productive Struggle

We need to help our students feel comfortable with struggle to instil persistence. Math is not about “being finished”. We are never finished! Cognitive science has shown that errors and misconceptions are essential to learning. Everyone will make errors, and they should be seen as a normal part of math learning.

Questions for reflection

- Do I model learning from errors?
- Do I demonstrate confidence and respect in my students’ abilities?
- Do I avoid praising a correct solution or displaying excitement over an interesting idea?
- Do I expect students to persevere to solve mathematical problems?
- Do I provide hints but not solutions?
- Do my responses encourage the students to keep thinking?
- Do I focus on the process, using a “wrong answer” as an opportunity to learn?

Peter Liljedahl’s research^{2} 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 establish a positive environment where students are encouraged to engage in struggle and collaborate, and creativity is both celebrated and required.

Interactions between students and teachers can impact student achievement. Meaningful feedback, mathematical mindset messages and positive relationships contribute to the success a child will experience. When these relationships include an opportunity to fail, an opportunity to struggle, an opportunity to contemplate, and 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.

## 4. Developing a 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.^{3} Research has shown that we consolidate mathematical ideas by communicating them.

In a math classroom the priority is problem solving, reasoning, and evaluation of mathematical situations (Small, 2010).

“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”^{4}

“Respect is more than tolerance and inclusion – it requires dialogue and collaboration.” 8 Ways: Aboriginal pedagogy from Western New South Wales. Dubbo, NSW, Australia: The Bangamalanha Centre (2012)

Questions for reflection

- 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?

^{1} Dagdag, J.M.H. and Dagdag, J. D. CONSTRUCTIVISM AND THE MATHEMATICS CLASSROOM ASSESSMENTS OF ELEMENTARY TEACHERS. (2020). Journal Of Critical Reviews, 7(12). doi: 10.31838/jcr.07.12.144 https://www.jcreview.com/fulltext/197-1592554034.pdf

^{2} Liljedahl, P. (2010). Peter Liljedahl. Retrieved 5 July 2020, from http://www.peterliljedahl.com/

^{3}Hattie, J., Fisher, D., Frey, N., Gojack, L., Delano Moore, S., & Mellman, W. (2016). Visible learning for mathematics, grades K-12. Corwin Publishers.

^{4}Williams, Chad. (n. d.) *Teaching and Understanding Elementary Mathematics: A companion document to the Saskatchewan Mathematics Curriculum*. Unpublished manuscript. (p. 24)

Share this