Academic journal article Australian Primary Mathematics Classroom

Where We Were ... Where We Are Heading: One Multiplicative Journey

Academic journal article Australian Primary Mathematics Classroom

Where We Were ... Where We Are Heading: One Multiplicative Journey

Article excerpt

A journey into multiplicative thinking by three teachers in a primary school is reported. A description of how the teachers learned to identify gaps in student knowledge is described along with how the teachers assisted students to connect multiplicative ideas in ways that make sense.

"We had to focus on understanding it ourselves and our school focus was multiplication and division. It needed to be more than an approach based on filling the gaps ... we had to focus on what we knew and start from there."

To introduce this 'journey', children in this Year Five class were working on a task that required them to match number sentences to word stories by cutting and pasting, and then to represent the story--this is an example of Bessie's work:

This may not necessarily be considered an amazing sample of a child's work but it represents some important development that took place in three classrooms in a primary school situated south of Perth.

This article documents part of the professional learning journey of three teachers at the school--Abbie, Carl, and Dan--and the extent to which their learning is reflected in the work of some of their students. It describes how these teachers have begun to develop into genuine 'connectionist teachers' who are well aware of how the 'big ideas' of mathematics are structured and related.


In 1997 Askew, Brown, Rhodes, Johnson and Wiliam wrote what has become a seminal text with regards to what characterises effective teachers of numeracy. They came up with three categories of teachers, with the most effective category given the title of connectionist teachers. One of the criteria that distinguished connectionist teachers was that they made rich connections between mathematical ideas. Unfortunately, the classroom being the busy place it is, is not always an environment which permits the time and space to reflect on how the 'bits' of mathematics fit together, and thereby allow teachers to develop the capacity to become connectionist. In fact, with the partitioned curricula that operate in most parts of the world it is a task beyond the majority of us. One way of connecting the mathematics is not to consider the atomised curriculum as individual bits of content but rather to consider what underpins mathematics --the 'big ideas'. Some authors (Charles, 2005; Hurst, 2014; Siemon, Bleckley & Neal, 2012) have taken the opportunity to reflect on this and become part of a conversation about what constitutes the big ideas of mathematics. All consider multiplicative thinking to qualify as a big idea as it underpins important mathematical concepts such as place value, division, fractions, measurement, statistical sampling, proportional reasoning, rates and ratios, and algebraic reasoning (Siemon, Beswick, Brady, Clark, Faragher, & Warren, 2011). Amongst other things, multiplicative thinking (MT) is about having a flexible understanding of a range of numbers and relationships between them, recognising and working with a range of multiplication and division situations, and communicating and understanding of these ideas in a variety of ways (Siemon, Breed, Dole, Izard, & Virgona, 2006).


In 2015, Abbie, Carl, and Dan's school undertook some professional learning (PL) regarding multiplicative thinking. This stimulated their interest to seek guidance as to how they could better develop MT with their students and how they could judge what elements of MT were already being well taught and learned in their school and which elements required further consolidation. A Multiplicative Thinking Quiz (MTQ) was administered with children in Years 4, 5, and 6 classes and results shared with the teachers. Teacher Abbie describes how this developed:

   This initial training and testing resulted in an

   increased awareness and interest by us into how
   these skills and understandings had an impact
   on the broader range of mathematical concepts
   taught through the primary school years. … 
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