4. Intelligent Practice
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Introduction

1. Introduction
Resource: my model of a Learning Episode
2. The benefits of online courses
3. Getting the most out of the course
4. Where are you at?
What I used to do

1. What my students' practice used to look like
2. Dodgy differentiation decisions
3. What do my students attend to?
Research: Patterns of variation in teaching the colour of light to Primary 3 students
4. Students cannot form expectations
Research: Variation and mathematical structure
Research: Errors Committed with High Confidence Are Hypercorrected
5. There's not a lot to discuss
6. A correct answer means thinking stops
7. The interesting maths is at the end
8. The importance of structured practice
What I do now

1. An example of Intelligent Practice
Activity: Product of prime factors (image)
Activity: Product of prime factors (pdf)
2. Analysing the product of prime factors sequence
Link: Product of prime factors (on website)
3. What do I mean by Intelligent Practice?
4. Where does Intelligent Practice fit in?
5. Purposeful Practice versus Intelligent Practice
6. Mathematical thinking
7. A tour of the site
Variation Theory
Key features

1. Different features of Intelligent Practice sequences
Activity: Volume with Pythagoras (image)
Activity: Volume with Pythagoras (pdf)
2. Confronting the obvious
Link: Volume with Pythagoras (website)
Activity: Rounding to multiples of (image)
Activity: Rounding to multiples of (pdf)
3. Confronting the unusual
Link: Rounding to the nearest multiple of (website)
Activity: Angles in isosceles triangles (image)
Activity: Angles in isosceles triangles (pdf)
4. An opportunity to generalise
Link: Angles in isosceles triangles (website)
Activity: Factorise into a single bracket (image)
Activity: Factorise into a single bracket (pdf)
5. Non-examples
Link: Factorise into a single bracket (website)
Activity: nth term of linear sequences (image)
Activity: nth term of linear sequences (pdf)
6. Interleaving high-value concepts
Link: nth term rule of linear sequences (website)
Reflect, Expect, Check, Explain

1. Reflect, Expect, Check, Explain to support mathematical thinking
Resource: The double pause of questioning
2. Reflect
Activity: Fractions of an amount - reverse (image)
Activity: Fractions of an amount - reverse (pdf)
Link: Fractions of an amount - reverse (website)
3. Expect
4. Check
5. Explain
Research: Self-explaining: The dual processes of generating inferences and repairing mental models
Research: The Self-Explanation Effect When Learning Mathematics: A Meta-Analysis
Research: Inducing Self-Explanation: a Meta-Analysis
Learning from Research: Worked-Out Examples: A Study on Individual Differences
Research: Self-explanations: How students study and use examples in learning to solve problems
Podcast: Ollie Lovell interviews Alexander Renkl
Model the first relationship

1. Reflect
Resource: my process for running Intelligent Practice questions in the classroom
2. Expect
3. Check
4. Explain
5. Model the first relationship summary
6. Division with fractions practice
7. Division with fractions analysis
Activity: Division with fractions (image)
Activity: Division with fractions (pdf)
Link: Division with fractions (website)
Student prompts

1. Mental addition practice
Activity: Mental addition (image)
Activity: Mental addition (pdf)
2. Mental addition anaylsis
Link: Mental addition (website)
3. Sample Space Diagrams practice
Activity: Sample space diagrams (image 1)
Activity: Sample space diagrams (image 2)
Activity: Sample space diagrams (pdf)
4. Sample Space Diagrams analysis
Link: Sample space diagrams (website)
5. A question...
6. Student prompts
Resource: Student prompts for Reflect, Expect. Check, Explain
Resource: RECE Student Prompt Sheet from George Stone
What do the students write down?

1. Plans and elevations practice
Activity: Plans and elevations (image)
Activity: Plans and elevations (pdf)
2. Plans and elevations analysis
Link: Plans and elevations (website)
3. Box plots practice
Activity: Box plots (image 1)
Activity: Box plots (image 2)
Activity: Box plots (pdf)
4. Box plots analysis
Link: Box plots (website)
5. A question...
6. Writing it down ideas
7. Too much writing
Resource: writing template
Resource: Intelligent Practice (example) by Nathan Day
Resource: Intelligent Practice (blank) by Nathan Day
Resource: Introducing RECE from @ah_haMaths
The 4-2 approach

1. Function machine practice
Activity: Function machines (image)
Activity: Function machines (pdf)
2. Function machines analysis
Link: Function machines (website)
3. Factorising practice
Activity: Factorise by grouping (image)
Activity: Factorise by grouping (pdf)
4. Factorising analysis
Link: Factorise by grouping (website)
5. A question...
6. Silent work
Research: Cortical Tracking of Speech-in-Noise Develops from Childhood to Adulthood
7. Paired discussions
Resource: Paired discussion prompts
8. The 4-2 approach
9. What do I do?
Discuss relationships

1. Probability of a single event practice
Activity: Probability of a single event (image 1)
Activity: Probability of a single event (image 2)
Activity: Probability of a single event (pdf)
2. Probability of a single event analysis
Link: Probability of a single event (website)
3. Median from a frequency table practice
Activity: Median from a frequency table (image 1)
Activity: Median from a frequency table (image 2)
Activity: Median from a frequency table (pdf)
4. Median from a frequency table analysis
Link: Median from a frequency table (website)
5. A question...
6. Discuss relationships
Resource: Discuss relationships prompts
Prompts for delving deeper

1. Combining ratio practice
Activity: Combining ratio (image)
Activity: Combining ratio (pdf)
2. Combining ratio analysis
Link: Combining ratio (website)
3. Multiplying and dividing terms practice
Activity: Multiplying and dividing terms (image)
Activity: Multiplying and dividing terms (pdf)
4. Multiplying and dividing terms analysis
Link: Multiplying and dividing terms (website)
5. A question...
6. Prompts for delving deeper
Resource: Prompts for delving deeper
Have we solved the problems?

1. Dodgy differentiation decisions
2. What do students attend to?
3. Cannot form expectations
4. There's nothing to discuss
5. A correct answer means thinking stops
6. The interesting maths is at the end
7. Mathematical thinking
Fill in the gaps

1. Straight line graphs practice
Activity: Fill in the gaps - straight line graphs (image 1)
Activity: Fill in the gaps - straight line graphs (image 2)
Activity: Fill in the gaps - straight line graphs (pdf)
2. Straight line graphs analysis
Link: Straight line graphs - fill in the gaps (link)
3. Fill in the gaps top tip
4. Time practice
Activity: Fill in the gaps - time (image 1)
Activity: Fill in the gaps - time (image 2)
Activity: Fill in the gaps - time (pdf)
5. Time analysis
Link: Time - fill in the gaps (website)
6. Fractions of an amount practice
Activity: Fill in the gaps - fractions of an amount (image)
Activity: Fill in the gaps - fractions of an amount (pdf)
7. Fractions of an amount analysis
Link: Fractions of an amount - fill in the gaps (website)
8. How to find Fill in the Gap activities
Atomisation

1. Coefficients and constants practice
Activity: Coefficients and constants (image)
Activity: Coefficients and constants (pdf)
2. Coefficients and constants analysis
Link: Coefficients and constants (website)
3. Choosing the correct trigonometric ratio practice
Activity: Trigonometry - which ratio (image)
Activity: Trigonometry - which ratio (pdf)
4. Choosing the correct trigonometric ratio analysis
Link: Trig - which ratio? (website)
5. Factorising into double brackets practice
Activity: Multiples to, adds to... (image)
Activity: Multiples to, adds to... (pdf)
Activity: Factorising into double brackets (image)
Activity: Factorising into double brackets (pdf)
6. Factorising into double brackets analysis
Link: Multiples to, adds to... (website)
Link: Factorising into double-brackets (website)
Fluency Practice

1. A question...
2. What is fluency?
3. The role of Fluency Practice
4. How do I make the decision?
5. What does Fluency Practice look like?
6. How much Fluency Practice do we need?
7. My favourite sources of Fluency Practice
Link: Maths Bot
Link: Corbett Maths
Link: CIMT MEP materials
Link: 10 ticks
Link: Practice makes perfect
Link: Maths4Everyone
Link: Increasingly difficult questions
Link: Maths HKO
Method Selection

1. A question...
2. Circles practice
Activity: Circles (image)
Activity: Circles (pdf)
3. Circle analysis
Link: Circles (website)
4. Expanding brackets practice
Activity: Expanding brackets (image)
Activity: Expanding brackets (pdf)
5. Expanding brackets analysis
Link: Expanding brackets (website)
6. Percentage change practice
Activity: Fill in the gaps - percentage change (image)
Activity: Fill in the gaps - percentage change (pdf)
7. Percentage change analysis
Link: Percentage change - fill in the gaps (website)
Useful links - from others

What is made possible to learn when using the variation theory of learning in teaching mathematics?
Variation: analysing and designing tasks
Teaching with Procedural Variation: A Chinese Way of Promoting Deep Understanding of Mathematics
Seeing an exercise as a single mathematical object: using variation to structure sense-making
Variation theory in mathematics education
NCETM CPD course on Variation
Podcast: Anne Watson and John Mason
Podcast: Anne Watson on the TES podcast
Blog: How to “teach” remotely
Variation Unplugged - with Anne Watson
Noticing and Attention - with John Mason
Conclusion

1. Where to next?
2. My online courses
Course certificate
Course feedback
Useful links - from me

My research paper collection
Book: How I wish I'd taught maths
Book: Reflect, Expect, Check, Explain
Mr Barton Maths website
Mr Barton Maths Podcast
Diagnostic Questions
Variation Theory
SSDD Problems
Maths Venns

Prompts for delving deeper

What are some of the challenges and probing questions we can ask our students to stimulate even deeper thinking?

13 Lessons