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Building an inclusive

community of inquiry

Hollie Walton

Teacher of mathematics, Cambridge

Andrew Blair

Head of mathematics, Ealing, London

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Inequalities in mathematics classrooms

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Inequalities in the maths classroom

Mathematics classrooms privilege some groups above others.

Gender

At year 9, boys performed significantly better than girls. The gender gap was as wide as in previous surveys and greater than the OECD average.
Significantly more girls than boys in years 5 and 9 lacked confidence in mathematics and did not like the subject, finding lessons dull, unenjoyable and off-putting.
Percentage of girls taking A-level Mathematics and Further Mathematics in 2019 was lower than in 2009 – girls 39% and 29% of entrants respectively.

PISA (2018) and TIMSS (2019)

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Inequalities in the maths classroom

Socio-economic status

Strongest indicator of achievement in mathematics. Students in years 5 and 9 in England who were eligible for FSM6 achieved scores significantly below the average of their peers and the gap widens as students get older.

Ethnicity

Black students are the lowest achieving group in both years 5 and 9. The results reproduce the pattern of inequitable achievement from previous rounds of the survey.

PISA (2018) and TIMSS (2019)

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Inequalities in the maths classroom

British Educational Research Journal (2019) The misallocation of students to academic sets in maths: A study of secondary schools in England

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Benefits of learning through inquiry

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Inquiry lessons

	‘Path smoothing’	Inquiry lessons
Learning	Discrete skills; small steps	Conceptual understanding and connections
Activity	Repetitive practice	Exploring, regulating and reflecting
Communicating	Teacher funnelling	Collaborative discussion
Thinking	Routine application	(Re)solving conjectures and questions

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Inquiry Maths Habits of Mind

Inquiry Maths

Poster

(Hamdi Ahmed)

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More inclusive classrooms

Learning mathematics through inquiry is more inclusionary than the traditional classroom setting.
Female students (Borasi, 1992)
Ethnic minority students (Hunter et al., 2020)

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Benefits of inquiry-based learning

Behaving like a mathematician.
Transferable skills.
Greater independence of learners.

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Starting an inquiry

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Inquiry Maths prompt

Diagram

Statement

Equation

Inquiry Maths

The sum of two fractions equals their product.

24 x 21 = 42 x 12

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Inquiry Maths

Promotes curiosity and questioning of the sort "Is it true that ...?" or "I've noticed ...". Ripe for speculation or conjecture. Stands just above the understanding of the class. Aimed at students' developing knowledge.

Fixed and changeable

Closed and open

Students must feel confident enough to manipulate and change the prompt; not intimidated by the prompt. Open enough to offer students the opportunity to regulate their own activity.

“Less to it and more in it”

Stripped back to the minimum, while simultaneously loaded with the potential for exploration.

Familiar and unfamiliar

Accessible and inaccessible

Obvious and intriguing

Inquiry Maths prompt

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Inquiry Maths prompt

40% of 70 = 70% of 40

50% of 10 = 10% of 50

47% of 74 = 74% of 47

40% of 30% of 20 = 20% of 30% of 40

Inquiry Maths

Alternatives

Stands just above the understanding of the class; aimed at students' developing knowledge.

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Starting an inquiry

Activity 1

Ask a question or make an observation about the prompt.

Inquiry Maths

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Inquiry Maths prompt

Inquiry Maths

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Orientation questioning and noticing

Inquiry Maths

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Inquiry Maths model

Inquiry Maths

Zuckerman: slow down

“Now the teacher must ‘slow down’ these sparks of curiosity and imagination so that they are registered and identified by other children, and other students become involved in the process of inquiry.� �Like thermal neutrons in a nuclear pile, this slowdown must provide for a self-perpetuating chain reaction of interactions in the class, propagating new questions that lead from the initial chaotic question to hypotheses that can be verified.”

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Regulatory cards

Inquiry Maths

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Regulatory cards

Inquiry Maths

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Agency Creating new cards

Inquiry Maths

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Including students’ ideas

4

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To decide

What questions to answer directly (‘arbitrary’ and ‘necessary’ knowledge)
What questions to keep open.
What to explore immediately.
What lines of inquiry to hold back.

The teacher’s response

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The teacher’s response

Inoue and Buczynski (2011) Stumbling blocks to inquiry

Unanticipated student response: Fails to anticipate students’ input and cannot give a pedagogically or mathematically meaningful response.
No student response: Fails to give a meaningful response to students’ silence or lack of input.
Disconnection from prior knowledge: Response severs connections between the lesson and students’ prior knowledge or their attempt to make sense of the concept.

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The teacher’s response

Inoue and Buczynski (2011) Stumbling blocks to inquiry

Lack of attention to student input: Ignores the students’ input.
Devaluing of student input: Diminishes student input by rejecting their suggestions and shuts down their attempts at making sense of the problem.
Mishandling of diverse responses: Does not know how to effectively manage or give meaningful ‘traffic controls’ to diverse responses to the prompt.

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Levels of inquiry

	Questions	Regulation	Pathways	Outcomes
Structured	Co-constructed	Restricted by the teacher
Guided	Student-led	Co-constructed
Open	Student-led	Student-led Teacher validation	Student-led Teacher instruction when required	Student-led Teacher assessed

Inquiry Maths

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Structured	New to inquiry; not easy to identify students who show curiosity or take initiative in maths lessons. Very few students, if any, are prepared to take risks.
Guided	Either new to inquiry, but contains an identifiable “breakthrough group” whose members generate ideas, offer conjectures and show high levels of curiosity, and have the influence to 'carry' their peers.�Or carried out inquiries before and the students are starting to suggest their own pathways.
Open	Experienced in inquiry; can take a prompt and inquire independently in order to create a mathematically-valid outcome (possibly without needing the regulatory cards).

Class profiles

Inquiry Maths

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5

Differentiation through lines of inquiry

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Inquiry Maths prompt

Inquiry Maths

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Line of inquiry Odds and evens

Inquiry Maths

First two expressions	Expression for the nth term of the intersecting sequence

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Line of inquiry Two numbers

Inquiry Maths

Esme and Olivia

(Year 9)

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Line of inquiry Other sequences

Inquiry Maths

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Line of inquiry Test conjectures

Inquiry Maths

Elijah and Tabassum

(Year 7)

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Line of inquiry Number of terms

Inquiry Maths

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Line of inquiry Diagrams

Inquiry Maths

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Line of inquiry Proof

Inquiry Maths

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Line of inquiry Further inquiry

Inquiry Maths

Chinese

Remainder

Theorem

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The teacher’s role in building an inclusive community of inquiry

6

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Becoming an inquiry teacher

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Developing norms and practices

Time and expertise required
Students’ prior experience of mathematics
Three strands - supporting contributions, establishing common ground and guiding the learning of mathematics.

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Developing norms and practices

Prompts to scaffold classroom talk

(Makar et al., 2015).

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Inquiry Maths and �Mixed Attainment Classes

Inquiry Maths

Devised and developed in mixed attainment classrooms.
Prompts promote learning at multiple levels.
Inquiry pathways involve students working on a common aim from different directions and at different levels of reasoning.
Students’ selection of an approach and mathematical level (guided by the teacher when necessary) ensures challenge and progress for all.
Inquiry unites the class in a mathematical process.
The unity of purpose promotes inclusiveness, cohesion and equity as all contributions add to the findings of the inquiry.