Measuring Messy Mathematics: Assessing learning in a mathematical inquiry context

The pedagogy of inquiry to teach mathematics presents a seemingly messy yet busy classroom, where children are engaged in using purposeful mathematics to collaboratively generate effective and creative solutions to open-ended, ambiguous questions. Problems arise when describing student learning in inquiry settings, when assessment practices are chosen that do not align with or are not designed to capture learning in this context. This study will present an inquiry into mathematics inquiry classrooms, to analyse and interpret the relationships between three classroom elements of assessment, teaching and learning. Distinctive to this study, the researcher was also the classroom teacher which offers the reader close insight into school practices in these classrooms. The researcher aimed to understand if an alignment between these classroom elements could support teaching and learning in inquiry settings. Using design research as a methodology (Cobb et al., 2003), two primary, inquiry classrooms (Years three and six) are presented as three iterative phases of study, each using particular theoretical lenses and analytical tools. In the first phase of study, theoretical analysis of formative assessment practices was based on Dewey’s (1891) conception of thinking as a process involving abstraction, comparison and synthesis. Vygotsky’s (1978) zone of proximal development was the analytical tool in the second phase of study and was used to analyse how the classroom teacher adjusted her teaching based on feedback she received through formative assessment. The third and final phase looked closer at the mathematical learning revealed through formative assessment using the DNR framework (Harel & Koichu, 2010): a Piagetian-influenced framework. Duckworth’s (2006) belief framework was also used to consider an interrelated synthesis of findings from this study, to assist in the development and refinement of assessment practices that align with using the inquiry pedagogy to teaching mathematics. Findings from three phases in this study revealed the inadequacy of summative assessment practices in capturing and describing student learning and thinking, fostered at higher levels through inquiry. In the first phase of study, analysis of assessment completed by students as part of their everyday, classroom curriculum reflected how such assessment only requires students to perform lower-level, reproductive thinking. In contrast, formative assessment opportunities encouraged students through inquiry to conceptualise their mathematical thinking in connected and abstract ways. The second phase of study focused on teaching in one inquiry classroom and characterised the difficulties classroom teachers face as they implement inquiry into their mathematics curriculum. Analysis of inquiry teaching and learning in this phase characterised how the teacher needed to be an engineer: able to interweave student ideas as potentialities, into the scaffolding of particular learning goals. Interweaving by the teacher, of students’ connections to the mathematical topic being explored, highlighted the complexity and messiness of the inquiry classroom where frequent interactions generated feedback about students’ thinking. Analysis of student learning in the third phase of study reflected a complex journey for students which considered interactions with peers in an inquiry context. Student thinking was provoked in these interactions shifting some responsibility for learning to the student as they tried to make sense of conflicting ideas. In all phases of this study, the inquiry pedagogy supported deep and connected mathematical learning, engineered by the classroom teacher towards particular learning and assessment goals. The learning process for students, as an ongoing journey of testing and refining mathematical processes and skills, was neglected when assessment did not value these characteristics. In inquiry, when assessment of learning values the messy and personal learning journey students face, there is potential for students to continue learning beyond the constraints of narrow curriculum objectives. Further research into ‘what else’ is learned through mathematics inquiry is required, to refine the pedagogy and to make its intentions clear. This study presents potential innovations to consider for future research

Reasoning with fractions: supporting pre-service teachers’ learning

We report on a mathematics education course for primary pre-service teachers (PSTs) in which both mathematical and pedagogical learning is essential. Part of this course is devoted to supporting PSTs in (a) strengthening conceptual understanding of fractions and (b) coming to imagine ways in which fractions can be effectively introduced in primary classrooms. In collaborative group work on fraction activities, PSTs generated solutions that were very different from those of year 3-5 pupils. This was unexpected and led to new insights and learning for the instructors. We draw implications about supporting the PSTs’ development of deeper mathematical reasoning in circumstances where prior mathematical learning limits their problem-solving skills and creativity. This work contributes to a larger effort of developing resources that could enhance adults’ learning across STEM disciplines

Don't just solve for x: letting kids explore real-world scenarios will keep them in maths class

In real life, ordering pizza for a group of people involves having a conversation about what people like, how much they can eat, how much they want to spend and whether pineapple really belongs on pizza.
 But in the context of a traditional maths class, the concept of ordering a pizza typically becomes a problem like this:
 If one pizza serves four children, how many pizzas do we need for a class of 28 children?
 An alarming number of Australian students don't choose mathematics in the senior school years. Figures from 2017 – the most recent available – show only 9.4% of Australian students in years 11 and 12 were enrolled in extended mathematics. This is the lowest percentage in more than 20 years

De novo CCND2 mutations leading to stabilization of cyclin D2 cause megalencephaly-polymicrogyria-polydactyly-hydrocephalus syndrome

Activating mutations in genes encoding phosphatidylinositol 3-kinase (PI3K)-AKT pathway components cause megalencephaly-polymicrogyria-polydactyly-hydrocephalus syndrome (MPPH, OMIM 603387). Here we report that individuals with MPPH lacking upstream PI3K-AKT pathway mutations carry de novo mutations in CCND2 (encoding cyclin D2) that are clustered around a residue that can be phosphorylated by glycogen synthase kinase 3β (GSK-3β). Mutant CCND2 was resistant to proteasomal degradation in vitro compared to wild-type CCND2. The PI3K-AKT pathway modulates GSK-3β activity, and cells from individuals with PIK3CA, PIK3R2 or AKT3 mutations showed similar CCND2 accumulation. CCND2 was expressed at higher levels in brains of mouse embryos expressing activated AKT3. In utero electroporation of mutant CCND2 into embryonic mouse brains produced more proliferating transfected progenitors and a smaller fraction of progenitors exiting the cell cycle compared to cells electroporated with wild-type CCND2. These observations suggest that cyclin D2 stabilization, caused by CCND2 mutation or PI3K-AKT activation, is a unifying mechanism in PI3K-AKT–related megalencephaly syndromes

Formative assessment tools for inquiry mathematics

Inquiry in mathematics is well suited to address authentic, ill-structured problems that are encountered in everyday life. However, available formative assessment tools are typically not designed for an inquiry approach. An exploratory study using Design Research aimed to understand and improve assessment practices of mathematical inquiry. Data collected from one classroom provided detailed examples of these assessment practices in action. Results from the initial stage and future directions of the project will be presented

Assessing for learning in inquiry mathematics

As pedagogy and assessment in primary mathematics classrooms move away from a focus on isolated facts, skills and procedures, pedagogy and assessment practices need to align with new ways of thinking and understanding. An inquiry approach to teaching mathematics incorporates open-ended, often ill-structured problems. Assessment, as part of this pedagogy, needs to be broad and flexible to capture construction of mathematical knowledge and understandings. This paper illustrates mathematical inquiry in practice and the assessment of related learning opportunities in a Year 3 (ages 7-8) classroom. Wiliam’s (2011) strategies for the effective use of assessment for learning were used as a framework to analyse these learning opportunities; the framework provided a useful tool to describe inquiry practices. Analysis of episodes from a primary classroom will offer evidence of learning and assessment practices in an inquiry mathematics unit, as a sequence of events over time

Introducing guided mathematical inquiry in the classroom : Complexities of developing norms of evidence

Guided Mathematical Inquiry (GMI) supports the development of deep understandings about mathematics concepts as students learn to address inquiry questions with evidence-based claims. Developing an evidence-based focus has been shown to be problematic with children. This paper presents how a framework for evidence was developed through the expert knowledge of teachers experienced in GMI with components trialled and illustrated using a Year 3 classroom unit on measurement and geometry. This framework can be used to give insight into the complexity of an evidence-focus in mathematics and support further research

Introducing Guided Mathematical Inquiry in the Classroom: Complexities of Developing Norms of Evidence

Guided Mathematical Inquiry (GMI) supports the development of deep understandings about mathematics concepts as students learn to address inquiry questions with evidence-based claims. Developing an evidence-based focus has been shown to be problematic with children. This paper presents how a framework for evidence was developed through the expert knowledge of teachers experienced in GMI with components trialled and illustrated using a Year 3 classroom unit on measurement and geometry. This framework can be used to give insight into the complexity of an evidence-focus in mathematics and support further research