1. Inquiry Maths and Mixed Attainment Classes Andrew Blair
Head of Mathematics, Haverstock School (Camden)
@inquirymaths
inquiry maths

2. Inquiry and mixed attainment1
inquiry maths

3. Inquiry Maths lessons Inquiry lessons Ofsted reports* Learning
Discrete skills
Conceptual understanding and connections
Activity
Repetitive practice
Regulating and reflecting
Communicating
Teacher funnelling
Collaborative discussion
Thinking
Routine application
(Re)solving conjectures and questions
* Understanding the Score 2008; Made to Measure 2012
inquiry maths

4. Inquiry Maths and Mixed Attainment Classes
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.
inquiry maths

5. Regulatory cards inquiry maths
After posing questions / making observation, students invited to participate in structuring / directing lesson. At same time cards suggest types of activities that are consistent with the discipline of mathematics (induction and deduction). Also include s ‘social’ cards on how to inquire. Change sets of cards depending on experience of class with inquiry (e.g. use the pack of 6 cards for less experienced inquirers).
inquiry maths

6. Regulatory cards inquiry maths
After posing questions / making observation, students invited to participate in structuring / directing lesson. At same time cards suggest types of activities that are consistent with the discipline of mathematics (induction and deduction). Also include s ‘social’ cards on how to inquire. Change sets of cards depending on experience of class with inquiry (e.g. use the pack of 6 cards for less experienced inquirers).
inquiry maths

7. Regulatory cards advantages
Self-differentiation: students choose point of access to the inquiry
Harmonisation of learning and concepts
Increases awareness of mathematical reasoning
Increases consciousness of own thinking
Increases constructive and critical agency
inquiry maths

8. Restricted by the teacher
Levels of inquiry
Questions
Regulation
Pathways
Outcomes
Structured
Co- constructed
Restricted by the teacher
Guided
Student-led
Open
Student-led teacher validation
Student-led teacher instruction when required
Student-led teacher assessed
(Optional)
inquiry maths

9. Introducing Inquiry Maths2
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10. What Inquiry Maths is not
Discovery learning (investigations)
Problem solving
Students’ everyday interests or a ‘real life’ context
Project-based learning
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11. Inquiry maths model Polya: deduction completes induction
We have discovered an interesting result but the reasoning that led to it was merely plausible, experimental, provisional, heuristic; let us try to establish it definitively by a rigorous proof. (How to Solve It, 1945)
The result of the mathematician’s creative work is demonstrative reasoning, a proof, but the proof is discovered by plausible reasoning, by guessing. (Mathematics and Plausible Reasoning, 1954)
Polya: deduction completes induction
IM aims to combine two forms of mathematical reasoning identified by Polya: “deduction completes induction”; procedures used and practised in service of exploration (e.g. 24 x 21 = 42 x 12). Differences between inquiry and investigation:
inquiry maths

12. Inquiry maths model Students learn to
Ask questions and notice properties
Make conjectures
Plan, monitor and reflect on their activity
Explore ideas in collaboration
Identify when they need new knowledge
Ask the teacher for instruction
Explain their reasoning
Prove their results
inquiry maths

13. Inquiry maths model Teachers aim to Connect concepts and procedures
Harness students’ curiosity
Connect concepts and procedures
Support student regulation
Co-construct open inquiries
Combine different forms of reasoning
Develop students’ initiative,
independence and leadership
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14. Starting an inquiry 2
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15. “Less to it and more in it.”
Inquiry Maths prompt
Diagram
Statement The sum of two fractions
equals their product.
Equation x 21 = 42 x 12
“Less to it and more in it.”
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16. Inquiry Maths prompt 40% of 70 = 70% of 40 50% of 10 = 10% of 50
Alternatives
Choosing a prompt, a teacher cannot just take one ‘off the shelf’ from the website. Should think about setting the prompt JUST ABOVE level of class, so involves a feature that is FAMILIAR (gives students confidence to question and observe) and UNFAMILIAR (intriguing). Example, one department with sets in year 8 changed the percentages prompt on the website to present the right balance of familiarity and intrigue for classes with different levels of attainment.
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17. Inquiry prompt
l + h = n 2 2
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18. Orientation questioning and noticing
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19. Regulatory cards
inquiry maths