“Math can be seamlessly integrated into children’s ongoing play and activities. But this usually requires a knowledgeable adult who creates a supportive environment and provides challenges, suggestions, tasks, and language.” (Sarama & Clements, 2009, pp. 332–333)
Once educators have a good understanding of the child and have developed their own mathematical knowledge for teaching, they can create situations which capitalize on children’s everyday mathematical knowledge. Researchers have identified five common core characteristics of early learning environments that support effective mathematical pedagogy and foster positive attitudes and beliefs about mathematics (Clements & Sarama, 2009, p. 259). The characteristics, which have been taken and/or adapted from this research, are described below.
The “Voices from the Field” belong to participants in the the Early Primary Collaborative Inquiry (EPCI), a ministry-supported network for primary educators, K to 2. Their examples and reflections, drawn from new professional learning and classroom experience, illustrate each characteristic in greater depth.
1. Use problems that have meaning for children (both practical and mathematical).
Context
By taking an inquiry stance, pose a balance of educator-initiated and child-initiated problems to individuals, small or large groups. Draw from students' intuitive mathematics and real-life experiences. Relate problems to the curriculum expectations and the mathematical processes. Challenge students with problems that are developmentally appropriate yet do not underestimate their abilities.
Voices from the Field
“ We began to value the mathematical processes rather than the final product and correct answer. We encouraged students to reason their way to a solution and the teacher's role shifted to facilitator.” “Our team began to pay more attention to what the students were genuinely interested in while they were playing and came up with real life problems to extend their thinking and to make things more purposeful.” “How we question students has changed. We see students as much more capable problem solvers.”
2. Expect that children will invent, explain and offer critiques of their own solution strategies within a social context.
Context
Provide opportunities for students to collaborate as they engage in mathematical activities so they can articulate, discuss and question each other’s thinking. Foster and encourage math talk and facilitate consolidation time with individuals, small or large groups. Value and honour a variety of strategies and solutions. Offer students regular opportunities to revise their thinking.
Voices from the Field
“We began to incorporate time for students to reflect and explain their thinking to other students as they solved these problems. This began to develop a collaborative culture in the classroom.” “Sometimes, we as educators, might assume that students do not have an understanding of mathematical concepts, when really,
they just did not have the mathematical language to explain their thinking.” “One practice that I have changed as a result of this inquiry is to bring students back together to share their solutions and thinking with their peers. This has made a big difference in the climate and culture in my classroom. It's a safer place to share our thinking.”
3. Provide opportunities for both creative invention and practice.
Context
Intentionally plan experiences based on observations of children engaged in activities. Provide a variety of materials so children can work through their strategies and make their thinking visible. Once a concept is acquired, provide practice experiences to consolidate learning. Practice is not meant to be rote or mechanical in nature; it occurs “through mathematical investigations that take place through free exploration, focused exploration, and guided activity” (Ontario Ministry of Education, 2010, p. 94). Sarama and Clements refer to this as mathematical play, or “playing with the math itself” (2009, p. 327).
Voices from the Field
“We used observations of children’s actions to understand and assess their reasoning.” “Focused observations helped to rethink the ability of many of the students. They proved to be more capable than expected.” “I have learned that paper and pencil tasks do not provide good, accurate assessment.” “How materials are presented makes as much difference as the choice of materials that are presented. The way that students use the materials is crucial to their learning."
4. Encourage and support children by providing carefully scaffolded opportunities which allow them to deepen their understanding, use meaningful and elegant solution strategies and confidently engage in the mathematics.
Context
Provide opportunities for students to mathematize their informal mathematics knowledge. Help students reflect on their strategies and representations. Nurture positive attitudes, self-efficacy, engagement and perseverance.
Voices from the Field
“We found out that really choosing questions carefully had an impact on students’ ability to deepen their understanding and express their ideas.” “We found that scaffolding in mathematics is more about the support we give children to solve a problem than about chunking problems into more manageable pieces.” “Unequivocally, the most noticeable impact on learning from involving students in inquiry was an increase in student engagement … students were on task, discussing the math during investigations, involved in problem solving, joyful and excited about what they were doing, and receptive to new learning from their peers.” “The three-part lesson gives children control and independence. They were participating and looking forward to being part of the process.”
5. Help children see connections between various types of knowledge and topics, with the goal of having each child build a well-structured, coherent knowledge of mathematics.
Context
Integrate mathematics activities into other subjects, like the arts and science, and make connections between the key mathematical ideas and the real world.
Voices from the Field
“A conscious effort was made to provide opportunities to reflect on student learning related to big ideas.” “We need to relate our problems to the real world and help the children make connections to the validity and usefulness of mathematics in their lives. Problem solving helps to develop curiosity, confidence and an open-mindedness that will help them to solve unfamiliar problems through their life and in the work force.”
Research reveals that young children come to school with a wealth of everyday mathematical knowledge that, if effectively built upon, can positively impact their future educational achievement. Knowledge about early mathematics learners and the mathematics for teaching can help educators create a rich environment and guide students to attain strong conceptual understandings, positive attitudes and self-efficacy. The research is also an excellent starting point for initiating conversations, at all levels of education, about the significance of early mathematics and how we can help all of our “splendid little mathematicians” develop the confidence and competence necessary to adeptly engage in mathematics throughout their lives.
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