5. The four ingredients of problem solving
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Introduction

1. Introduction.mp4
2. The benefits of online courses
3. Getting the most out of the course
4. Where are you at?
Three types of problems students struggle with

1. The contextual problem
2. The subtle contextual problem
3. The multi-topic problem
4. What about other types of problems?
Some controversial statements about problem solving

1. Problem solving and learning
Research: Consequences of history-cued and means-end strategies in problem solving
Blog: The Difference Between Solving a Problem and Learning Some Math From It
2. When to do problem solving
Podcast: Andrew Blair interview from 2017
Podcast: Andrew Blair interview from 2014
Link: Inquiry maths website
Link: Kapur's section on Productive Failure
Research: Element interactivity as a factor influencing the effectiveness of worked example–problem solving and problem solving–worked example sequences
Blog: John Sweller on Productive Failure
3. The two year rule
Article: The fundamental problem with teaching problem solving
4. Problem solving strategies
Podcast: Michael Pershan on problem solving strategies
Routines

The importance of routines.mp4
Podcast: Bruno Reddy on routines
Ingredient 1 - SSDD Problems

1. What is the problem?
Research: The Benefit of Interleaved Mathematics Practice is not limited to Superficially Similar kinds of Problems
2. What is the solution?
Link: SSDD Problems website
3. Why might SSDD Problems be a good idea?
4. Some of my favourite SSDD Problems
5. Narrow the domain (method select)
6. The SSDD Problems website.mp4
7. SSDD Problems with different age groups
8. SSDD Problems in different subjects
9. SSDD Problems in the classroom
10. What do students say?
11. Idea: Mixed-topic starters
Research: Using Spacing to Enhance Diverse Forms of Learning
12. Idea: Command words
A teacher's guide to command words
13. Idea: Problem solving for all
14. Idea: Group work
15. Idea: High-value skills
16. Idea: DSID
Link: Miss Konstantine on Twitter
17. Idea: Planning for error
18. Idea: Providing a purpose
19. Idea: Reacting to exam questions
20. Idea: (Blind) modeling
21. Idea: What is the question?
22. Final reflection
Resource: SSDD Problems (blank) by Nathan Day
Resource: SSDD Problems (example) by Nathan Day
Ingredient 2 - Goal-Free Problems

1. What is the problem?
Research: Cognitive Load Theory, Learning Difficulty and Instructional Design
Blog: The goal-free approach: My talk from CogSciSci 2019
2. What is the solution?
Blog: Sweller on the difference between Goal-Free Problems and minimally guided instruction
3. Goal-Free Problems with different age groups
4. The Goal-Free Problems website
Link: The Goal-Free Problems website
5. But they won't look like this in the exam
6. Goal-Free Problems in the classroom
7. Idea: In lessons
Link: White Rose Maths problems of the day
8. Idea: Exam preparation
9. Idea: Exam technique
Blog: John Sweller on exam technqiue
10. Idea: Goal-Free mock exams
11. Idea: Problem of the week
12. Idea: When to use Goal-Free Problems
13. Reflection
Ingredient 3 - Purposeful practice

1. What is the problem?
Book: The hidden lives of learners
2. What is the solution?
Activity: Simplifying ratio Venn Diagram 1
Activity: Simplifying ratio Venn Diagram 2
3. Principles of Purposeful Practice
Research: Developing mathematical fluency: comparing exercises and rich tasks
4. The power of Purposeful Practice
5. Purposeful Practive versus Intelligent Practice
6. Venn Diagrams - laying the foundations
Link: CIMT MEP materials
Link: Maths Venns website
7. Venn Diagrams - prompts for thinking deeper
Prompts for thinking deeper
8. Venn Diagrams - in the classroom
Activity: Types of number Venn Diagram
9. Venn Diagram - averages and range
Activity: Averages and range Venn Diagram
10. Venn Diagrams - straight line graphs
Activity: Straight line graphs Venn Diagram
Venn Diagrams on the playground
11. Venn Diagrams - time to move on
12. Colin Foster - linear equations
Link: Mathematical etudes website
Activity: Expression polygons
13. Colin Foster - enlargement
Activity: Staying on the page
Resource: Geogebra file for enlargements
14. Don Steward - written subtraction
Link: Median maths blog
Activity: Two-digit subtraction
15. Don Steward - factors
Activity: Four factors
Activity: Factor facts
16. Open Middle - FPD conversions
Link: Open Middle website
Activity: Converting between fractions and decimals
17. Open Middle - area of quadrilaterals
Activity: Area of a quadrilateral
18. NRICH - adding fractions
Link: NRICH website
Activity: Almost one
19. NRICH - expanding and simplifying
Activity: The simple life
20. Reflection
Ingredient 4 - UKMT questions

1. Introduction
Link: UKMT website
2. UKMT Problems in the classroom
3. Source - Diagnostic Questions
Link: UKMT questions on Diagnostic Questions
Resource: UKMT (example) by Nathan Day
Resource: UKMT questions (blank) by Nathan Day
4. Source - Dr Frost Maths
Link: Dr Frost Maths
5. Source - NRICH short problems
Link: NRICH short problems
6. Reflection
Useful links - from others

10 part blog series: How do kids think about solving equations?
Research: Developing students’ strategies for problem solving in mathematics
Video: Lessons for Mathematical Problem Solving
Video: Alan Schoenfeld on Poyla's problem solving approach
Blog: Problem solving or explicit instruction: Which should go first when element interactivity is high? by Greg Ashman
Conclusion

1. Where to next.mp4
2. My online courses.mp4
Course feedback
Course certificate
My online courses
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

2. What is the solution?

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