How can we make the practice we give our students as effective as possible?
My choice of the practice questions I give my students has changed dramatically in recent years. In my model of a Learning Episode, such practice is the bridge between modelling and problem solving. As such, it must solve a problem I have had for many years with the kind of practice questions I used to give my students... how on earth do I get differentiation right? In this course we dive into why I believe that carefully designed sequences of questions, together with the process of Reflect, Expect, Check, Explain, may be the solution. They allow students to gain valuable practice, whilst also providing opportunities to think mathematically. In this course I model the actions, prompts and structure I use to enable my students to have the best chance of benefiting from such an approach, as well as look at examples from schools all around the world who have made this process work for them. We also look at what Fluency Practice looks like and when we might choose to give it to our students, the role of Intelligent Practice during Atomisation, and why Method Selection is so important following both Intelligent and Fluency Practice.
Please note: this course formed the basis of my book: Reflect, Expect, Check, Explain: sequences and behaviour to enable mathematical thinking in the classroom, but here I use brand-new sequences of questions and examples that are not featured in the book, as well as new ideas I have learned from schools who have read the book and tried out the ideas for themselves. Therefore the course should be useful whether you have read the book or not.
Please see the bottom of the page for FAQs about suitability, cost, payment options, and more.
Feedback from the end of course survey
I now have a much deeper understanding about how your questions on Variation Theory work and how it is supposed to look in the classroom - looking forward to making the links between questions more explicit.
I'm sold on your approach - it's kind of everything I've intuitively believed and it's so inspiring to now have a guiding framework for it and a shared vocabulary for it. I feel like I can't thank you enough because your work is so helpful in so many ways!
What did the practice I used to give my students look like, and why was it problematic?
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
What kind of practice do I now give my students?
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
What are some of the key features of intelligent practice sequences of questions that I like to include?
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
What is the mathematical behaviour that underpins intelligent practice?
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
What does the initial modelling phase look like in the classroom?
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
What support prompts are available to support students?
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?
What are some of the challenges students face when writing down explanations, and how can we support them?
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
How do I balance the benefits of silent work and collaboration?
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
How do I go through the answers and bring the sequence together?
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
What are some of the challenges and probing questions we can ask our students to stimulate even deeper thinking?
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?
We return to the issues presented at the start of the course to see if we have resolved them
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
What does this activity structure look like and how can we get the most out of it?
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
What role does intelligent practice play in the Atomisation section of the Learning Episode?
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
Where does fluency practice fit into all of this?
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
What is method selection, why is it so important, and how can intelligent practice help?
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
A selection of links you might find useful
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
A few things before we say goodbye
1. Where to next?
2. My online courses
Course certificate
Course feedback
Useful links - from me
Links to some of my other work
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
FAQs
Is this course suitable for primary school teachers?
I am a secondary school maths teacher by training, and I make no claim whatsoever to have any expertise in the domain on primary teaching. However, I have been lucky enough to run this Intelligent Practice course many times with primary colleagues, and it seems to go down well. I believe that the key principles are transferable, and I try my best to use a wide variety of examples that are suitable for all age groups. The challenge - as it is with everyone who takes the course - will be to think hard about what they would need to do to make the ideas work for them and their students.
Is this course suitable for non-maths teachers?
I don't think so. Every example is from the world of maths, and I have not yet seen any successful implementation of the core ideas in another subject. I believe if is possible, but I wouldn't like to make any claims that this course will be useful or the ideas will be directly transferable to other domains.
Is this course suitable for non-UK teachers?
Yes! I have been lucky enough to work with teachers all around the world, and I ensure wherever possible that my courses are not tied to any specific curriculum or specification. I am confident that aside form my weird accent, teachers from other countries will find most of the ideas relevant and transferable.
How long can I access the course content for?
As long as this platform exists! That is one of the key advantages of an online course - you can go back over the content again and again.
If, for whatever reason, the platform shuts down or I need to remove content, I will give you as much notice as possible (I will aim for at least 6 months) so you can complete the courses.
In addition, from time to time I will update the course content with new videos, resources and ideas. I will email you when this happens and you will have access to this as well for no extra cost.
Can I pay with an invoice instead of online?
The easiest way to pay is online. The service accepts all major cards as well as PayPal. Paying this way gives you immediate access to the course.
But if you need to pay via invoice, then no problem! Just send an email to mrbartonmaths@gmail.com with details of:
1. The email addresses of the delegates taking the course
2. Your school name and address
3. Who to email to invoice to
Then I will send you an invoice and register your colleagues on the course
Can I get a VAT receipt?
Of course!
If you have paid online, just login, click on the drop-down menu next to your picture on the top-right of the screen, select Billing and you can print off your VAT receipt(s) there.
If you pay-offline (by emailing me as described above) then I will email you an invoice which will serve as your VAT receipt.
Can I buy one pass and then share it with my colleagues?
I am afraid not. The price of each course is per person.
Each person who pays for the course has their own log-in details, so the platform can keep track of their individual progress. This allows you to log back on using any device and pick up where you left off.
Are there discounts available?
If you want to purchase a bundle of passes for the courses - perhaps you have a large department or you are part of an Academy chain - send me an email telling me what you have in mind, and hopefully we can reach a deal!
