Creating learning spaces that foster creative thinking in mathematics

Kristjana Skúladóttir, mathematics teacher

Abstract

Children’s thinking about mathematics and understanding how they learn and engage with mathematics has gained more importance in the teacher education programs and the national curriculum in Iceland. This article provides a case study of the work of one mathematics teacher Kristjana, whose goal was to engage her elementary students (ages 6–13) in mathematics through problem-solving and creative thinking. She has also provided professional development to teachers and has been a valued mentor to many teachers in their journey towards more engaging mathematics teaching. Kristjana is an active member of Flötur, the Association of Mathematics Teachers in Iceland. This article highlights the research Kristjana has been introduced to and how she has used her knowledge about children’s thinking to guide her instructional practice and support other teachers. We used narrative inquiry to tell Kristjana’s story, using evidence from Kristjana’s career, interviews with Kristjana, field notes, and sample work from Kristjana’s students. Kristjana’s story reveals how her collaborations with mathematics education researchers and colleagues have informed her journey. Through ongoing discussions about the importance of observing children’s thinking, Kristjana and her colleagues collaborated to design a learning environment that supports student mathematical thinking. In this chapter, we will highlight three bodies of research that have influenced Kristjana: Cognitively Guided Instruction (CGI), research on student’s proportional reasoning, and assessment for learning.

Cognitively Guided Instructions has had a profound impact on Kristjana since she was introduced to it in the early 1990s. Around that time, Dr Thomas Carpenter and Dr Elizabeth Fennema introduced CGI to the mathematics education community. The premise of CGI is that children have informal knowledge of mathematics, which should be the base of their formal learning in primary school mathematics. Specifically, CGI investigated the informal knowledge children used to solve simple word problems based on whole number operations prior to being taught how to solve them. Based on the student strategy analysis, they developed a framework for different problem types for addition, subtraction, multiplication, and division problems and showed that students’ intuitive strategies developed from direct modeling to more abstract strategies as they formalised their knowledge of the four operations and their understanding of base-ten concepts. In practice, understanding children’s thinking is essential to design learning environments that put the child in the forefront and where children’s learning is based on their way of knowing.

After being introduced to CGI, Kristjana participated in the Making Meaning of Proportion project, which was grounded within the ideas of CGI that students generate strategies based on what they know. This classroom-based research project explored the development of students’ understanding of proportion. The big idea underlying proportion is two multiplicative relationships: the ratio of one quantity to another and the ratio of a scaled version of the quantities to the original version. The premise of the project was to create a learning environment where students’ thinking was center stage as they engaged with these essential ideas of proportion through solving problems. The project showed that the strategies students generate to solve missing value proportion problems were primarily based on the number of relationships in a problem, as well as the problem context and students’ understanding of proportion. Two broad levels of correct strategies for missing value problems were identified: build-up strategies at different efficiency levels and multiplicative strategies. During instruction, the specific numbers used in problems could be manipulated to engage students with the two multiplicative relationships in a proportion. Kristjana’s participation as one of the teachers in the project expanded her knowledge of CGI and gave her the understanding to apply the CGI approach to the more complex mathematical domain of proportional reasoning.

Instructional practices based on assessment for learning was the third large body of work that influenced Kristjana’s teaching. Instead of using assessment to measure student performance, formative assessment uses assessment tools that promote student progress. The national curriculum in Iceland emphasizes formative assessment and assessment for learning. In the same spirit, it states that teachers should strive to find out what students know, and assessments should be designed in such a way that students can demonstrate their understanding. Students should reflect on their own learning to help them reach their own learning goals. The focus of formative assessment is on the students, using assessment to provide feedback to students and to support them to become responsible for their own learning. Kristjana’s beliefs and teaching practices harmonize with the ideas of CGI, proportional reasoning, and assessment for learning. Using children’s thinking and assessment supports teachers in talking with their students about the problems responding to their ideas, and building on them when planning their lessons