By Michael Telek
Institute for Learning
Dr. Kristen Tripet has spent more than a decade in and around the classroom trying to discover an answer to a question she found herself wrestling with as a young student, ‘Is there a better way to teach mathematics?’
Tripet, an IFL partner currently serving as program manager for reSolve: Maths by Inquiry in Canberra, Australia, was recently awarded the MERGA (Mathematics Education Research Group of Australasia) Early Career Research Award for a research paper, The Reification of the Array: The Case of Multi-digit Multiplication, which was based on her 2019 doctoral thesis.
“For me, there was a lot of information around in early multiplication understanding, though it’s very limited in the multi-digit multiplication. So that’s where that interest came from,” said Dr. Tripet. “My focus was looking at a learning trajectory for multi-digit multiplication, in terms of a trajectory for a two-week teaching sequence. That was the focus of my work with a real focus on how students’ strategies developed, like them reinventing conventional mathematics for themselves.”
The crux of her research, involving 45 fifth grade students in Sydney, was looking at representations and how students use them. During this study, students would be using an array to understand a problem (in this case it was a bakery fulfilling cupcake orders) before the teacher would slowly move to more abstract versions of the array, and eventually an area model.
“What I found was that to actually make sense of the mathematics, kids kept going back to an earlier version [of the array],” said Dr. Tripet. “As they were making sense of the mathematics, they removed themselves from the context of the problem. I think that was significant contribution to the research that was out there.”
Another big takeaway according to Dr. Tripet was looking at the students’ cognitive development and the social development of understanding. By examining these perspectives, Dr. Tripet found herself asking how do you support the individual and how do you support the community of learners in the classroom?
That’s now a question reSolve and the IFL are looking to uncover. While these organizations search for a solution when it comes to students, they are also focusing on communities of inquiry in terms of teachers. Dr. Tripet said she will be doing some research into assisting groups of teachers with the goal of adjusting pedagogical practices while acknowledging the practical knowledge educators bring into the classroom and appreciating teacher agency.
Conclusions from Dr. Tripet’s Work related to the Reification of the Array
- Students need the opportunity to use the array to explore the multiplicative structure, at first making connections between specific situations and the arrays that model them, and then using arrays for more generalized mathematical reasoning.
- Students should not be restricted to one form of the array. Different forms of the array will serve different functions for students depending on their current levels of understanding. Students need opportunities to select and use a form of the array that makes sense to them as they make sense of and reason about the multiplicative structure.
- Mapping the process of how understanding of arrays evolves, provides guidance to practitioners–teachers and coaches, alike–and offers insight into the design of instructional materials.
With Australian education officials preparing a new mathematics curriculum set to debut in 2022, there will be a lot of work needed to support teachers implementing the new curriculum and new resources to go along side it. That includes Content-Focused-Coaching from the IFL. Dr. Tripet says the reSolve program in collaboration with the Department of Education in New South Wales will soon begin training math coaches to eventually become facilitators of the Content-Focused-Coaching model.
“[Content-Focused-Coaching] is fantastic. This is such rich professional learning that’s embedded in the classrooms with teachers actually working on changing practice,” said Dr. Tripet. “One of the biggest things with the reSolve project, rather than talking now about professional learning, I’ve shifted to talking about professional learning and support. It’s got to be about supporting the classroom. Any professional learning has to be about them supporting teachers in in their classrooms to implement change.”
The reSolve has seen such success with Content-Focused-Coaching in math, they are now working to adopt the work within the Science curriculum.
“I’m having a real love for the work that comes out of the IFL. One of the things I will highly recommend are the Taking Action1 books. Using that as a basis for communities of inquiries, I think it can be really, really useful resource for teachers,” said Dr. Tripet.
1 The Taking Action: Implementing Effective Mathematics Teaching Practices series, published by the National Council of Teachers of Mathematics (NCTM), offers a coherent set of grade-specific professional learning experiences designed to foster teachers’ understanding of eight effective mathematics teaching and practices and their ability to apply those practices in their own classrooms.