A Piagetian tool to support the development of teaching

Written by Alan Edmiston

Although Piaget did most of his testing on his own children, and indeed those of other Professors in Geneva, I do think his work has something to offer those in the mathematics classroom today. In this article I will give an example of how his ideas supported a colleague and I to make some important decisions concerning a low achieving Year 9 group just before lockdown. The material used was developed as part of my an Erasmus plus- funded project, called ACTS (Assessment Companion for Thinking Skills), a partnership project between University of Lincoln, Let’s Think Forum, Turku University in Finland TA Group in Latvia.

As part of the project, I developed a hierarchy of thinking competency in the key maths strands of algebra, data manipulation and ratio.  To do this I used some of the ideas behind the CAME (Cognitive Acceleration through Math’s Education) project, developed by Michael Shayer, David Johnson and Mundher Adhami, and their analysis of the descriptors of thinking as supplied by Piaget. This article introduces the algebra assessment hierarchy, and explains how teachers can use it to

The algebra tool attempts to show increasing maturity/sophistication in thinking based upon what students might say or do when working on a problem that requires the use of algebraic understanding. It is aimed at students from the age of nine upwards and breaks down the Piagetian descriptors of concrete and formal thinking into four stages or levels. At its highest level, formal operational thinking, it contains all of the features of good understanding that any successful student at post-16 would naturally exhibit. As an assessment tool it is concerned not with what students know, or can recall, but with the quality and sophistication of their reasoning.

The algebra tool, shown below, seeks to provide as many descriptors of algebraic thinking as necessary to enable anyone observing a group of students to be able to assess the quality of their thinking. When I was trying out the tool many teachers used it to sit with a quiet child whose mathematical thinking they felt unsure of, and to listen to them working on a task. There is only so much we can find out from written assessments and so I think any resource that allows teachers to make detailed observations of students’ thinking can only be a good thing.

As part of the refinement phase I asked a subject lead for mathematics if he could use the tool to assess a class while I taught them a CAME lesson.  We decided to do this with a low achieving Year 9 group. and the lesson I chose was lesson 21 which is called Expressions and Equations, and which focuses upon the difference between expression and equations. The advantage is that the start is very concrete and allows all students to participate whilst the latter phase is challenging enough to allow children who could achieve a grade 4 and 5 to be challenged.

What quickly emerged, after about 15 minutes of starting the lesson, was that only four pupils out of the 18 in the group were able to progress beyond the first episode of the lesson. Once they began working on the first main task it became clear that the pupils struggled to work with expressions, and particularly this question: “What is the largest value that 3x + 15 could have?”

The expected answer is that it can be infinite but the ideas of:

  • 3x meaning 3 lots of x and,
  • adding a fixed value to a variable,

really threw the majority of the class.  Luckily, three students in the class had some useful insights into the problem. The lesson provided an opportunity for these students to support the rest of the class and develop their understanding.  As they spoke, they grew in confidence so my role as teacher was to provide the opportunity for them to express and clarify their thinking.

Following the lesson, the teacher and I discussed the fact only the four students described above had demonstrated any of the skills listed in the top half of the tool. The head of maths felt this was a consequence of a poor diet of learning experiences at the start of these students’ secondary maths career. This in turn highlighted the need for him and his department to provide opportunities in Years 7 and 8 for pupils to reason and to make their mathematical thinking open to discussion with their peers.

The above episode provides an example of how the tool can be used to support teachers’ and students’ learning. You could use the tool to listen to a class working on a problem that requires them to use algebra. For example, sit in on a colleague’s lesson, identify what you hear and then discuss the implications for learning. Compare Year 7 with Year 10 in a task that requires the pupils to think and reason, and share findings with colleagues. Whatever you decide to do with the tool, please get in touch to share the outcomes and your thinking: [email protected]