Craig Barton interviews guests from the wonderful world of education about their approaches to teaching, educational research and more. All show notes, resources and videos here: https://www.mrbartonmaths.com/blog/

Kris Boulton returns to the podcast to discuss atomisation and how it can lead to unstoppable learning for our students. You can access the show-notes here: mrbartonmaths.com/blog/atomisation-kris-boulton

Timestamps:

- Atomization in mathematics education, with a focus on expertise-induced blindness and its impact on students' understanding. (10:57)
- Breaking down complex processes into simpler steps. (15:12)
- Teaching methods using a taxonomy of seven concepts (categories, comparative, transformation, fact, and process). (19:40)
- Categorical concepts, comparatives, and transformations in mathematics. (23:43)
- Identifying and teaching mathematical ”atoms” for better instruction. (29:38)
- Teaching math concepts by breaking them down into smaller, familiar ”atoms” to help students understand and build upon them. (38:18)
- Simplifying math expressions using factoring and atomization. (44:09)
- Teaching math concepts by breaking them down into smaller, more manageable ”atoms” to help students understand and build confidence. (49:48)
- Categorical concepts in math education, with a focus on non-examples. (53:43)
- Importance of examples in teaching, with a focus on the limitations of language in conveying concepts. (57:31)
- Teaching concepts using examples and definitions. (1:03:31)
- Using correlated features in math teaching. (1:08:50)
- Teaching quadratics with examples and caveats. (1:13:28)
- Using examples to teach concepts, including minimally different and maximally different examples. (1:19:20)
- Teaching language learners using negative examples first. (1:27:01)
- Teaching math concepts using examples and testing sequences. (1:31:20)
- Decisions and categorization in math education. (1:35:10)
- Using language to make math problems easier. (1:39:51)
- Differentiating between cognitive routines and transformations in math. (1:45:04)
- Teaching math concepts using different methods. (1:48:25)
- Teaching math concepts to children using a step-by-step approach. (1:54:18)
- Using mini whiteboards for testing sequences in math class. (1:59:19)
- Teaching strategies, emphasizing the importance of interactive learning and using whiteboards. (2:02:36)
- Using simplified symbols vs. expert-level symbols in math education. (2:07:59)
- Using continuous conversion in math lessons. (2:12:00)
- Teaching math concepts using cognitive routines. (2:20:02)
- Teaching math concepts to students using explicit and implicit methods. (2:25:28)
- Teaching strategies, including non-examples, identifying concepts, and managing classroom noise. (2:32:03)
- Math education, examples, and training. (2:35:28)
- Improving math education with technology and hybrid learning models. (2:39:19)
- Teaching methods and classroom management. (2:43:42)
- Teaching math to mixed-ability students, emphasizing the importance of exploration and unveiling mathematical concepts. (2:50:08)
- Teaching math to high school students, focusing on approach for different learners. (2:53:42)
- Teaching probability with creative problem-solving strategies. (2:57:47)
- Breaking down complex math concepts into smaller parts for better understanding. (3:01:57)
- Sequencing examples in teaching, emphasizing clarity and brevity. (3:06:53)
- Using ”atomization” to teach math concepts more efficiently. (3:11:09)

**Sponsor details**

This episode of the Mr Barton Maths podcast is proudly supported by Casio Education. Check out the fantastic resources they have available for teachers here: education.casio.co.uk/mr-barton-maths

**Episode details**

In this conversation, Craig Barton and Kris Boulton discuss the concept of atomization in teaching mathematics. Atomization involves breaking down complex routines into smaller, teachable components called atoms. The atoms can be categorized into categories, comparatives, transformations, and facts. The goal is to make every component of a routine familiar to students so that they can focus on how the components fit together. Teachers can improve their ability to identify atoms by practicing and reflecting on their own teaching. Atomization allows for more effective and efficient teaching of mathematics. The conversation explores the concept of atomization in teaching math, specifically focusing on the use of categorical concepts as atoms. The importance of non-examples in helping students understand categories is highlighted. The conversation also discusses the mechanics of using examples and non-examples, as well as the role of language in teaching concepts. The concept of correlated features as a type of atom is introduced. The conversation concludes with reflections on the effectiveness of different teaching approaches. In this part of the conversation, Kristopher Boulton and Craig Barton discuss the importance of using minimally different and maximally different examples in teaching. They explain how to construct sequences that help students generalize concepts and make decisions. They also explore the distinction between transformations and categories, and the role of cognitive routines in instruction. In this part of the conversation, Kristopher and Craig discuss the concept of continuous conversion and the importance of overtization in teaching mathematics. They explore the idea of using mini whiteboards to facilitate data collection and the power of using examples that require minimal changes. They also discuss the structure of cognitive routines and the importance of allowing students to figure things out on their own. In this final part of the conversation, Kristopher Boulton discusses his teaching approach for simultaneous equations and the importance of backward chaining. He also explores how to help students remember the steps involved in solving simultaneous equations and the role of narration in instruction. Boulton emphasizes the need for training and centralized resources to implement his approach effectively. He also discusses the challenges of atomization in non-routine questions and the potential benefits of technology in education. Boulton concludes by addressing the issue of what goes in books versus mini whiteboards and the applicability of his approach to higher achieving students.

**Takeaways**

- Atomization involves breaking down complex routines into smaller, teachable components called atoms.
- Atoms can be categorized into categories, comparatives, transformations, and facts.
- The goal is to make every component of a routine familiar to students so that they can focus on how the components fit together.
- Teachers can improve their ability to identify atoms by practicing and reflecting on their own teaching.
- Atomization allows for more effective and efficient teaching of mathematics. Atomization involves breaking down complex concepts into smaller, manageable parts called atoms.
- Non-examples are important in helping students understand categories by highlighting what does not belong.
- The use of examples and non-examples should be carefully balanced to avoid overwhelming students.
- Language plays a crucial role in teaching concepts, but it should be used sparingly and only when necessary.
- Correlated features can be used as a type of atom to help students understand relationships between concepts. Using minimally different and maximally different examples helps students generalize concepts and make decisions.
- Teaching sequences should include negative examples, minimally different positive examples, and maximally different positive examples.
- Transformations involve infinite responses, while categories involve finite responses.
- Cognitive routines are higher-order transformations that can be broken down into lower-order transformations.
- The method used to teach a concept does not determine the process of atomization. Continuous conversion involves turning one example into the next, allowing students to see the changes and the impact they have.
- Overtization is the process of making hidden or covert information explicit and visible to students.
- Using mini whiteboards can facilitate quick data collection and allow students to make minimal changes to their work.
- Cognitive routines should focus on allowing students to figure things out on their own, rather than relying on the teacher to provide all the answers.
- The structure of cognitive routines can vary, but it is important to provide clear instructions and allow students to engage in problem-solving. Backward chaining can be an effective strategy for teaching simultaneous equations.
- Narrating the steps can help students remember the order of chains in solving equations.
- Training and centralized resources are crucial for implementing Boulton’s teaching approach.
- Atomization may present challenges in non-routine questions, but the approach can still be applied.
- Technology can play a role in enhancing education and individualizing learning.
- The choice between books and mini whiteboards depends on the nature of the task and the need for monitoring and feedback.
- Boulton’s approach is beneficial for higher achieving students as it promotes deeper understanding and flexibility in mathematics.

**Time-stamps:**

- Atomization in mathematics education, with a focus on expertise-induced blindness and its impact on students’ understanding. (10:57)
- Breaking down complex processes into simpler steps. (15:12)
- Teaching methods using a taxonomy of seven concepts (categories, comparative, transformation, fact, and process). (19:40)
- Categorical concepts, comparatives, and transformations in mathematics. (23:43)
- Identifying and teaching mathematical “atoms” for better instruction. (29:38)
- Teaching math concepts by breaking them down into smaller, familiar “atoms” to help students understand and build upon them. (38:18)
- Simplifying math expressions using factoring and atomization. (44:09)
- Teaching math concepts by breaking them down into smaller, more manageable “atoms” to help students understand and build confidence. (49:48)
- Categorical concepts in math education, with a focus on non-examples. (53:43)
- Importance of examples in teaching, with a focus on the limitations of language in conveying concepts. (57:31)
- Teaching concepts using examples and definitions. (1:03:31)
- Using correlated features in math teaching. (1:08:50)
- Teaching quadratics with examples and caveats. (1:13:28)
- Using examples to teach concepts, including minimally different and maximally different examples. (1:19:20)
- Teaching language learners using negative examples first. (1:27:01)
- Teaching math concepts using examples and testing sequences. (1:31:20)
- Decisions and categorization in math education. (1:35:10)
- Using language to make math problems easier. (1:39:51)
- Differentiating between cognitive routines and transformations in math. (1:45:04)
- Teaching math concepts using different methods. (1:48:25)
- Teaching math concepts to children using a step-by-step approach. (1:54:18)
- Using mini whiteboards for testing sequences in math class. (1:59:19)
- Teaching strategies, emphasizing the importance of interactive learning and using whiteboards. (2:02:36)
- Using simplified symbols vs. expert-level symbols in math education. (2:07:59)
- Using continuous conversion in math lessons. (2:12:00)
- Teaching math concepts using cognitive routines. (2:20:02)
- Teaching math concepts to students using explicit and implicit methods. (2:25:28)
- Teaching strategies, including non-examples, identifying concepts, and managing classroom noise. (2:32:03)
- Math education, examples, and training. (2:35:28)
- Improving math education with technology and hybrid learning models. (2:39:19)
- Teaching methods and classroom management. (2:43:42)
- Teaching math to mixed-ability students, emphasizing the importance of exploration and unveiling mathematical concepts. (2:50:08)
- Teaching math to high school students, focusing on approach for different learners. (2:53:42)
- Teaching probability with creative problem-solving strategies. (2:57:47)
- Breaking down complex math concepts into smaller parts for better understanding. (3:01:57)
- Sequencing examples in teaching, emphasizing clarity and brevity. (3:06:53)
- Using “atomization” to teach math concepts more efficiently. (3:11:09)

**Soundbites:**

- “You want everything the kids see to feel familiar.”
- “When you come to executing that chain of atoms from start to finish, nothing, not one thing should be new there anymore.”
- “Atomization involves breaking down complex routines into smaller, teachable components called atoms.”
- “Something is transformed and what was difficult and complicated and hard suddenly becomes easy.”
- “The teacher did something that not enough teachers do… He’d actually assessed prerequisite knowledge.”
- “The kids feel so smart doing this. So smart.”
- “I think your inclusion of, I think you’re treating this as a correlated feature sequence and saying this is or this isn’t because it’s probably the right intuition.”
- “When you should have minimally different examples, and for the audience we’re still talking in the realms of categories here, so whether something is in this group or it isn’t in this group, like the quadratic example, the prism example and so on, so when should examples be minimally different, and when should they be maximally different, and why do I love this circle diagram so much?”
- “The snappier name is negative first.”
- “I’ve aimed for as little speaking as possible”
- “Silently add the different bits, leave it for a second or two and then point and say because 2x plus 5x is 7x, 3y plus 10y is 13y and 10 plus 5 is 15”
- “The expansion sequence is changing the form”
- “I was teaching my year nine class mix attainment to solve simultaneous equations.”
- “I think that active narration is probably helpful.”
- “I’ve written level one to intentionally try to make it not very daunting.”

**Resources from this episode:**

- To keep in touch with Kris and his Unstoppable Learning project, click here

**More about Kris:**

- On Twitter Kris is @Kris_Boulton
- His excellent blog can be found at: tothereal.wordpress.com
- My first interview with Kris is here
- My second interview is here
- My third interview is here

**My usual plugs:**

- You can help support the podcast (and get an interactive transcript of all new episodes) via my Patreon page at patreon.com/mrbartonmaths
- If you are interested in sponsoring an episode of the show, then please visit this page
- You can sign up for my free Tips for Teachers newsletter and my free Eedi newsletter
- My online courses are here: craigbarton.podia.com
- My books are “Tips for Teachers“, “Reflect, Expect, Check, Explain” and “How I wish I’d taught maths”