# Teacher Development for Equitable Mathematics Classrooms: Reflecting on Experience in the Context of Performativity

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## Abstract

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## 1. Introduction and Background

## 2. Pathways to Equitable Mathematics Classrooms

… expanding mathematical competence does not mean watering down mathematics. Indeed, the opposite is true: Expanding mathematical competence means rendering authentic mathematical smartnesses both visible and consequential in classrooms. Looking at mathematics as a field, we see that its great accomplishments have not come about from quick and accurate calculations, but from other kinds of insights, creativity, and intelligence: asking good questions, making astute connections, working systematically, seeing patterns, illustrating representations, and so on ([6] p. 36)

## 3. Teacher Learning in a Climate of Performativity

## 4. Reflections on Teacher Learning: Telling the Story

## 5. Moving towards Equitable Mathematics Classrooms—Identifying Key Events

#### 5.1. Questioning Traditional Teaching

#### 5.2. Learning to Listen to Students

#### 5.3. Recognising the Need for Informal Mathematics

Aidan is teaching a bottom set Year 8 group of 10 students. The first question in his quick quiz starter is presented as the multiplication 16 × 12 written above a rectangle accurately drawn 12 down and 16 across. The side lengths are not labelled, and the outline is drawn on a faintly squared blackboard. The students set to work drawing the rectangle on squared paper, and one enquires if it needs to be ‘dead accurate’. Aidan circulates to gain a sense of student approaches. After six minutes, he stops the group and talks to them about how to behave when observing each other working at the board. He reminds them to use manners, be respectful, and remember that they are on a journey to becoming mathematicians. He talks about ‘us’ and ‘we’, and that we are looking for shortcuts, spotting patterns, making connections, and recognising that we are all at different stages of that journey.

Aidan invites a student to come to the board to show how he found the number of squares. Beginning with the bottom left-hand square, he touch-counts each square, writing ‘1’ in the bottom square through to ‘12’ at the top of the column (see Figure 1). The rest of the group watch. The counting is slow. To me, it feels slightly tedious but apparently not to the class and certainly not to Aidan who comments on the precision with which the student is counting and the hard work that it takes to work in this way. Another student comments that ‘you could have just written in 12 because it says it’s 12 down the side’. The first student, seemingly not ready to make this connection, continues counting down the second column of squares, writing in numbers as he goes, “13. 14. 15. 16. 17. 18. 19.”. Aidan instructs him to stop there and asks ‘Where is it going to end? What is the last number he will write in this column?’. He reminds the class not to shout out in order to allow thinking time. After a 30 s pause, Aidan accepts a student’s suggestion of 24 with the response ‘How did you know it was going to be 24?’. The student refers to 1 and 1 making 2, 2 and 2 making 4. Aidan responds with ‘Ok’ and moves on to hear from another student. We return to the student at the board and repeat the sequence of counting, stopping part way and hearing other students’ rationale for what goes in the last square of the third column. Aidan provides a meta-commentary on their strategies, noting that they are pattern spotting. He carefully selects who comes to the board, based on the strategy they have used. The next student numbers only the bottom square of each row to reveal a total of 192 squares. A third partitions the original rectangle into 4 smaller sized rectangles, but her partitioning does not match the 100/60/20/12 totals she writes in each mini rectangle. Their methods are all based on counting the squares inside although some have developed short-cut ways to do this. None of the class are attempting to multiply 16 × 12 using a standard algorithm. Eighteen minutes of the lesson is given to sharing strategies, with Aidan commenting on them and directing some students to try others’ strategies when it comes to the next lesson.

#### 5.4. Developing Teaching That Values Informal Understanding

#### 5.4.1. Intervening in Student Thinking—The Area Project

**Figure 6.**Struggling to fit same-sized squares when the image is not drawn to scale: a student strategy from the Year 9 post-test.

#### 5.4.2. Encountering RME: Redefining Progress

#### 5.4.3. Connecting Informal and Formal

#### 5.5. Using My Learning in Teacher Education

## 6. Discussion: Developing and Delivering Professional Development for Equity in the Current Education Climate

## 7. Implications for Future Research on Professional Development

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Touch-counting strategy used to find the total number of squares of a 16 × 12 rectangle. Reproduction of classroom board work.

**Figure 3.**Calculating perimeter rather than area: a common student strategy from the Year 9 pre-test.

**Figure 4.**Adapting the drawing in squares method for a right-angled triangle: a student strategy from the Year 9 post-test.

**Figure 5.**Adapting the drawing in squares method for a parallelogram: a student strategy from the Year 9 post-test.

**Figure 8.**Drawing in squares and counting up in multiples of 12: student solution strategy from the Year 9 post-test.

**Figure 9.**Partitioning the height dimension only and reasoning why multiplication will find the total number of squares: student solution strategy from the Year 9 post-test.

Aim | Activity |
---|---|

To develop teacher awareness and knowledge of students’ natural informal approaches | Teachers study student solutions to mathematics problems, analysing approaches, looking for connections across methods, linking student answers to how they are taught, ranking solutions from informal to formal |

To enable teachers to experience RME both as a teacher and as a learner of mathematics | Trainers model RME lessons in which teachers are positioned as students and trainers provide meta-commentary on their pedagogic decisions. Key strategies: bring learners to the board to showcase a range of their solutions; remain neutral; focus learners on solutions with directions such as ‘Say what you see’, ‘Can you draw something?’, ‘What’s the same, what’s different?’ [about these solutions]. |

To develop teachers’ appreciation of how RME uses context and models to build a different view of progress | Focus on how a particular model, such as the bar, emerges from many contexts to become a model that learners can apply elsewhere across many topic areas, even to non-contextual, bare number questions |

To provide models of teaching that shift the role of teacher from transmission orientation to that of a facilitator | Video observation emphasising noticing and accurate, non-evaluative description. Focus on, e.g., what neutral teacher responses look like and what teachers see as mathematical ‘progression’ |

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**MDPI and ACS Style**

Hough, S.; Solomon, Y.
Teacher Development for Equitable Mathematics Classrooms: Reflecting on Experience in the Context of Performativity. *Educ. Sci.* **2023**, *13*, 993.
https://doi.org/10.3390/educsci13100993

**AMA Style**

Hough S, Solomon Y.
Teacher Development for Equitable Mathematics Classrooms: Reflecting on Experience in the Context of Performativity. *Education Sciences*. 2023; 13(10):993.
https://doi.org/10.3390/educsci13100993

**Chicago/Turabian Style**

Hough, Sue, and Yvette Solomon.
2023. "Teacher Development for Equitable Mathematics Classrooms: Reflecting on Experience in the Context of Performativity" *Education Sciences* 13, no. 10: 993.
https://doi.org/10.3390/educsci13100993