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How I Wish I Had Taught Maths: Reflections on research, conversations with experts, and 12 years of mistakes
"I genuinely believe I have never taught mathematics better, and my students have never learned more. I just wish I had known all of this twelve years ago."Craig Barton is one of the UK's most respected teachers of mathematics. In his remarkable new book, he explains how he has delved into the world of academic research and emerged with a range of simple, practical, effective strategies that anyone can employ to save time and energy and have a positive impact on the long-term learning and enjoyment of students.
Craig presents the findings of over 100 books and research articles from the fields of Cognitive Science, Memory, Psychology and Behavioural Economics, together with the conversations he has had with world renowned educational experts on his Mr Barton Maths Podcast, and subsequent experiments with my students and colleagues.
Craig presents the findings of over 100 books and research articles from the fields of Cognitive Science, Memory, Psychology and Behavioural Economics, together with the conversations he has had with world renowned educational experts on his Mr Barton Maths Podcast, and subsequent experiments with my students and colleagues.
452 pages, Paperback
Published May 16, 2023
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Displaying 1 - 30 of 34 reviews
September 14, 2018
Great book based on research, rather than subjective intuition. Although I feel that most of his counterintuitive results were actually things that already feel like the best way of doing things. His initial (and he argues wrong) approach of letting students struggle and find things for themselves always seemed impractical to me and suited only for experts.
Here is my personal summary of things that I liked, which I wrote while reading:
-----
Diagnostic Questions:
• Use often to diagnose required knowledge for current class - can be very quick
• Should be <10 sec to think,
• Should test only one skill (not many),
• Clear & unambiguous
• Impossible to get right with misconceptions
• Each wrong answer must correspond to a certain misconception, so it gives us diagnostic information.
• This teachers uses fingers, one for A, two for B, .... leaves students the time to think and then goes one,two,three vote! Students vote in silence. Then in order asks those for A, then B, then C, etc to explain their reason. Then does a re-vote. Giving the correct answer only after the re-vote. Rub out incorrect answer to avoid misconceptions.
Example-Problem pairs. "Worked example" followed by "Your turn" (this teacher adds the extra of: "Supercharged" problem, if in the middle you ask students to reflect, forcing them to Self-Explain).
• Worked example done in silence first. Then narrating what I did. Students not copying so as to have full attention. Then let students copy it. Then their turn to do a sample problem IDENTICAL to the example (but changing numbers).
• When doing a sequence of examples, use MINIMAL VARIATION between one and the next. Like an experimental method, changing only one thing to see its effect. Also examples seem a coherent whole rather than a random selection of examples. Make slight change so that students predict result. If its wrong, they will remember it much better because they had predicted it.
• Make sure examples include boundary examples! Quirky or unusual examples that will help students eliminate under-generalization and overgeneralization. Throw students off autopilot. Forces students to self-explain to themselves.
• Make sure no ambiguity in the question. Make sure there are not different (wrong) ways to arrive at the correct answer.
• Make sure no ambiguity is allowed in student answers, e.g. draw parallelogram, what if they draw a square? Either they are masters or know nothing, it's difficult to tell!
Deliberate practice:
• Like musicians practising separate bits of piece rather than whole piece at a time, and footballers practising individual skills rather than whole matches. Identify substeps that a novice sees in a single expert step. Practice the sub-steps INDIVIDUALLY and isolated from each other, to master each and all of them. No risk in having questions look very different to final exams. Giving final exam style questions might overload students
• Analogy: struggling to solve a problem (eg. changing a tire for the first time) can be so slow, confusing, and demanding of short term memory, that even if finally done successfully, next time will still be difficult. Learning is much less efficient than if you had a proper led training with deliberate practice of the steps to automate them.
Purposeful Practice (how to avoid boring repetition of similar quick problems):
• Purposeful Practice: lies between a boring series of exercises and a complex problem whose choice of method is uncertain. It is something which requires students to perform the specific task to be practiced (no choice of method) many times, but with an ultimate goal or purpose which makes it feel different. E.g instead of giving a list of fractions to add, give a set of fractions and ask students to find ways to add them to get the closest possible to 1. Novice learners will just practice the procedure of adding fractions while experts will learn new things, new ways of thinking.
Problems with early gains (my label) tips:
• Make a problem in which students are asked to find out everything they can, rather than a final single aim. This helps students feel relaxed, gives early gains and motivation as things are being found out.
Surface vs Deep structure in problems:
• Problems have surface structure (e.g a school trip... or a triangle is...) and deep structure (eg averages, Pythagoras, areas, perimeters). It is most common to batch problems with different surface but same deep structure (DS-SD) but we should also introduce problems with same surface but different deep structures! (SS-DD) so that students learn to recognise the differences in deep structures! Eg present problems that are all on triangles but relying in completely different things: areas, perimeters, ratios, angles, Pythagoras, ... all looking identical.
Errors: hypercorrection effect
• Errors committed with high confidence are more likely to be reflected on and learnt. Realising errors committed with low confidence does not shake the mind as much and has less likely a learning effect. So before showing answers to students, ask them to rate their confidence!
Long term memory
• Students might show amazing performance on the day you teach. Next week it’s as if they knew nothing.
• Retrieval power (ease to remember right now) vs Storage power (long term ease of increasing retrieval power).
○ Studying increases both.
○ If retrieval power is too high, then the act of retrieving will not increase storage power.
○ The lower the retrieval power, the highest increase in storage power when successfully retrieved.
Therefore, students need time to begin forgetting before retrieving!
This has been identified and replicated, "one of the most general and robust findings of experimental research on memory"
The spacing effect
• Dedicate part of each lesson to review concepts from weeks earlier
• Homework should be mostly for earlier topics (author recommends 2/3) and only 1/3 for current topic!
• Teacher’s exams and quizzes should be cumulative
Interleaving of topics
• Having to resolve through interference of topics Forces learners to find similarities and differences and higher level learning.
• Try to have current questions include as much earlier material as possible
Here is my personal summary of things that I liked, which I wrote while reading:
-----
Diagnostic Questions:
• Use often to diagnose required knowledge for current class - can be very quick
• Should be <10 sec to think,
• Should test only one skill (not many),
• Clear & unambiguous
• Impossible to get right with misconceptions
• Each wrong answer must correspond to a certain misconception, so it gives us diagnostic information.
• This teachers uses fingers, one for A, two for B, .... leaves students the time to think and then goes one,two,three vote! Students vote in silence. Then in order asks those for A, then B, then C, etc to explain their reason. Then does a re-vote. Giving the correct answer only after the re-vote. Rub out incorrect answer to avoid misconceptions.
Example-Problem pairs. "Worked example" followed by "Your turn" (this teacher adds the extra of: "Supercharged" problem, if in the middle you ask students to reflect, forcing them to Self-Explain).
• Worked example done in silence first. Then narrating what I did. Students not copying so as to have full attention. Then let students copy it. Then their turn to do a sample problem IDENTICAL to the example (but changing numbers).
• When doing a sequence of examples, use MINIMAL VARIATION between one and the next. Like an experimental method, changing only one thing to see its effect. Also examples seem a coherent whole rather than a random selection of examples. Make slight change so that students predict result. If its wrong, they will remember it much better because they had predicted it.
• Make sure examples include boundary examples! Quirky or unusual examples that will help students eliminate under-generalization and overgeneralization. Throw students off autopilot. Forces students to self-explain to themselves.
• Make sure no ambiguity in the question. Make sure there are not different (wrong) ways to arrive at the correct answer.
• Make sure no ambiguity is allowed in student answers, e.g. draw parallelogram, what if they draw a square? Either they are masters or know nothing, it's difficult to tell!
Deliberate practice:
• Like musicians practising separate bits of piece rather than whole piece at a time, and footballers practising individual skills rather than whole matches. Identify substeps that a novice sees in a single expert step. Practice the sub-steps INDIVIDUALLY and isolated from each other, to master each and all of them. No risk in having questions look very different to final exams. Giving final exam style questions might overload students
• Analogy: struggling to solve a problem (eg. changing a tire for the first time) can be so slow, confusing, and demanding of short term memory, that even if finally done successfully, next time will still be difficult. Learning is much less efficient than if you had a proper led training with deliberate practice of the steps to automate them.
Purposeful Practice (how to avoid boring repetition of similar quick problems):
• Purposeful Practice: lies between a boring series of exercises and a complex problem whose choice of method is uncertain. It is something which requires students to perform the specific task to be practiced (no choice of method) many times, but with an ultimate goal or purpose which makes it feel different. E.g instead of giving a list of fractions to add, give a set of fractions and ask students to find ways to add them to get the closest possible to 1. Novice learners will just practice the procedure of adding fractions while experts will learn new things, new ways of thinking.
Problems with early gains (my label) tips:
• Make a problem in which students are asked to find out everything they can, rather than a final single aim. This helps students feel relaxed, gives early gains and motivation as things are being found out.
Surface vs Deep structure in problems:
• Problems have surface structure (e.g a school trip... or a triangle is...) and deep structure (eg averages, Pythagoras, areas, perimeters). It is most common to batch problems with different surface but same deep structure (DS-SD) but we should also introduce problems with same surface but different deep structures! (SS-DD) so that students learn to recognise the differences in deep structures! Eg present problems that are all on triangles but relying in completely different things: areas, perimeters, ratios, angles, Pythagoras, ... all looking identical.
Errors: hypercorrection effect
• Errors committed with high confidence are more likely to be reflected on and learnt. Realising errors committed with low confidence does not shake the mind as much and has less likely a learning effect. So before showing answers to students, ask them to rate their confidence!
Long term memory
• Students might show amazing performance on the day you teach. Next week it’s as if they knew nothing.
• Retrieval power (ease to remember right now) vs Storage power (long term ease of increasing retrieval power).
○ Studying increases both.
○ If retrieval power is too high, then the act of retrieving will not increase storage power.
○ The lower the retrieval power, the highest increase in storage power when successfully retrieved.
Therefore, students need time to begin forgetting before retrieving!
This has been identified and replicated, "one of the most general and robust findings of experimental research on memory"
The spacing effect
• Dedicate part of each lesson to review concepts from weeks earlier
• Homework should be mostly for earlier topics (author recommends 2/3) and only 1/3 for current topic!
• Teacher’s exams and quizzes should be cumulative
Interleaving of topics
• Having to resolve through interference of topics Forces learners to find similarities and differences and higher level learning.
• Try to have current questions include as much earlier material as possible
May 21, 2020
Wow, this is great!
Like Craig Barton I've also taught math for many years so this brought back many memoriess.
Here's some of what I believed as a teacher:
(1) Open ended class discussion are really good.
(2) Teaching procedures without trying to explain the theory behind them is really bad.
(3) (Most) Students can only be motivated by grades (extrinsic reward).
(4) If students are struggling to understand or solve a question, that's a good sign (no pain, no gain!)
I think the fundamental delusion I was laboring under is what I'll call "the expert fallacy": We teachers often assume that students are just like us. Those assumptions I listed above actually do work pretty well for the most capable, well prepared and motivated math students. These are the kind of students who would enthusiastically contribute to the class discussion, eagerly absorb the theory and enjoy competing for the highest grade.
But they don't work nearly as well for the average student, who probably has a shaky mathematical background and is just struggling to connect this new material with what they already know.
And I think that's where my teaching failed, because I didn't properly consider what would be the best techniques for this type of student.
Three main categories of things I took away from this book:
(1) Experts think differently from beginner's
Experts see a topic they are familiar with in "chunks". We see the entire structure. To someone seeing it for the first time, they see it as disconnected so it's harder for them to process. Therefore:
(i) Discovery based learning, might not work as well for a first introduction to a topic because the back and forth discussion might just confuse the students who are trying to grasp the basics.
(ii) It might be a good idea to teach the basic algorithm and procedures without explaining them. Familiarity with the procedure will bread confidence, and they can get the explanation for why it works later.
(iii) It could be a good idea to break down a multi-step procedure into its separate parts and practice them separately (even if the individual parts are not independently useful). This reduces the cognitive load on the students and will make it easier for them to do the entire procedure.
(iv) Doing diagnostic testing before starting each topic to see what major misconceptions students have relative to that topic can be really helpful.
(2) Motivation
(i) I now realize that I underestimated the possibilities of intrinsic motivation. But the way the naïve students gets excited might be different from the mathematician. The mathematician might get excited about the abstract mathematical beauty, but the student's excitement might be more similar to a kid's excitement when they figure out how to make a new toy work (nothing wrong with that of course!). We underutilize the opportunity to elicit this kind of excitement because we are so focused on explaining "why" something works.
(ii) Success leads to motivation. When students feel they understand a topic and they are doing well with it, they are much more motivated to keep studying it and keep going. Failure tends to lead to discouragement and giving up. Unfortunately I think teachers can be quick to write off students who as "lazy and unmotivated" when if they had been able to have some early success they might have stayed motivated and learned a lot more.
(iii) I liked this simple general trick for motivating a topic:
-Have students solve a problem in an easy but very inefficient way.
-Then teach them the better way.
They will automatically appreciate why the better way is so cool.
(3) Testing and Remembering
(i) Testing for misconceptions - Deliberately designing questions to identify specific misconceptions is a simple but brilliant idea that, to my knowledge, is very little applied.
(ii) Spaced repetition - He gives a few variations of this, but the basic idea of systematically reviewing each topic a few weeks after it has been taught seems very easy to implement and potentially quite powerful, especially if you are covering a large number of very unrelated topics.
(iii) Don't always give detailed feedback - He makes a good argument that in many cases, it's better not to give very detailed feedback. If the student has to think about why they got the question wrong they will actually learn more.
The only criticism I have is that he gives so many interesting ideas it's hard to see how you could implement them all. But that's a good problem to have.
This will definitely significantly influence the next class that I teach.
Like Craig Barton I've also taught math for many years so this brought back many memoriess.
Here's some of what I believed as a teacher:
(1) Open ended class discussion are really good.
(2) Teaching procedures without trying to explain the theory behind them is really bad.
(3) (Most) Students can only be motivated by grades (extrinsic reward).
(4) If students are struggling to understand or solve a question, that's a good sign (no pain, no gain!)
I think the fundamental delusion I was laboring under is what I'll call "the expert fallacy": We teachers often assume that students are just like us. Those assumptions I listed above actually do work pretty well for the most capable, well prepared and motivated math students. These are the kind of students who would enthusiastically contribute to the class discussion, eagerly absorb the theory and enjoy competing for the highest grade.
But they don't work nearly as well for the average student, who probably has a shaky mathematical background and is just struggling to connect this new material with what they already know.
And I think that's where my teaching failed, because I didn't properly consider what would be the best techniques for this type of student.
Three main categories of things I took away from this book:
(1) Experts think differently from beginner's
Experts see a topic they are familiar with in "chunks". We see the entire structure. To someone seeing it for the first time, they see it as disconnected so it's harder for them to process. Therefore:
(i) Discovery based learning, might not work as well for a first introduction to a topic because the back and forth discussion might just confuse the students who are trying to grasp the basics.
(ii) It might be a good idea to teach the basic algorithm and procedures without explaining them. Familiarity with the procedure will bread confidence, and they can get the explanation for why it works later.
(iii) It could be a good idea to break down a multi-step procedure into its separate parts and practice them separately (even if the individual parts are not independently useful). This reduces the cognitive load on the students and will make it easier for them to do the entire procedure.
(iv) Doing diagnostic testing before starting each topic to see what major misconceptions students have relative to that topic can be really helpful.
(2) Motivation
(i) I now realize that I underestimated the possibilities of intrinsic motivation. But the way the naïve students gets excited might be different from the mathematician. The mathematician might get excited about the abstract mathematical beauty, but the student's excitement might be more similar to a kid's excitement when they figure out how to make a new toy work (nothing wrong with that of course!). We underutilize the opportunity to elicit this kind of excitement because we are so focused on explaining "why" something works.
(ii) Success leads to motivation. When students feel they understand a topic and they are doing well with it, they are much more motivated to keep studying it and keep going. Failure tends to lead to discouragement and giving up. Unfortunately I think teachers can be quick to write off students who as "lazy and unmotivated" when if they had been able to have some early success they might have stayed motivated and learned a lot more.
(iii) I liked this simple general trick for motivating a topic:
-Have students solve a problem in an easy but very inefficient way.
-Then teach them the better way.
They will automatically appreciate why the better way is so cool.
(3) Testing and Remembering
(i) Testing for misconceptions - Deliberately designing questions to identify specific misconceptions is a simple but brilliant idea that, to my knowledge, is very little applied.
(ii) Spaced repetition - He gives a few variations of this, but the basic idea of systematically reviewing each topic a few weeks after it has been taught seems very easy to implement and potentially quite powerful, especially if you are covering a large number of very unrelated topics.
(iii) Don't always give detailed feedback - He makes a good argument that in many cases, it's better not to give very detailed feedback. If the student has to think about why they got the question wrong they will actually learn more.
The only criticism I have is that he gives so many interesting ideas it's hard to see how you could implement them all. But that's a good problem to have.
This will definitely significantly influence the next class that I teach.
April 5, 2018
Phenomenal. If there ever was a Bible for Maths teachers this is it! Great mixture of research and practical application. Craig writes in a very winsome and humble way. Love the summaries at the conclusion of each chapter, very helpful.
January 6, 2019
I wish this book had been written seven years ago when I started teaching. A fact-filled journey of discovery. Thought it focuses on math teachers, there are lessons here that can be applies to any subject. A round-up of research. The idea of how it the research changed Barton's lessons impacted me as a classroom teacher.
Reading this was a challenge, it is not a page turner, but it makes finishing it more rewarding in the end.
Reading this was a challenge, it is not a page turner, but it makes finishing it more rewarding in the end.
July 1, 2020
This book is an absolute triumph.
It is a warm and often humorous account of a dedicated teacher’s journey to master their craft, full of practical ideas. Honestly, as an RQT I found it quite inspiring. It is steeped in the most recent and thorough research, yet it wears it’s references lightly, never feeling like name dropping and the academic theories are laid out clearly with fantastic practical ideas, explaining how they look and can work well in the classroom.
I’m a Primary teacher, I teach Maths 25% of the week, and work on the English team at school, yet I found this as helpful as any other book I’ve read, as the principles and ideas can be applied across the curriculum.
It’s also a treasure trove of suggestions for books, research and bloggers to look into further so I can’t wait to start exploring the references even more.
If you’re interested in teaching and/or the (re)emerging theories of knowledge, explicit instruction and spaced practice, as well as many others, or just want to read a witty, enjoyable and thorough book on our captivating profession, then I couldn’t recommend it enough.
It is a warm and often humorous account of a dedicated teacher’s journey to master their craft, full of practical ideas. Honestly, as an RQT I found it quite inspiring. It is steeped in the most recent and thorough research, yet it wears it’s references lightly, never feeling like name dropping and the academic theories are laid out clearly with fantastic practical ideas, explaining how they look and can work well in the classroom.
I’m a Primary teacher, I teach Maths 25% of the week, and work on the English team at school, yet I found this as helpful as any other book I’ve read, as the principles and ideas can be applied across the curriculum.
It’s also a treasure trove of suggestions for books, research and bloggers to look into further so I can’t wait to start exploring the references even more.
If you’re interested in teaching and/or the (re)emerging theories of knowledge, explicit instruction and spaced practice, as well as many others, or just want to read a witty, enjoyable and thorough book on our captivating profession, then I couldn’t recommend it enough.
August 20, 2019
I have taken so much from this book that I look forward to putting in practice. Unlike other reviewers, I am a new maths teacher and this book will be the foundation of my practice. I wonder how different my teaching would be without it. The three best things about this book are:
1) The structure: each section is broken into several sections that make it very easy to follow. I also really appreciate the comprehensive references and the summary sections at end of each section. The chapters (each made up of several sections) also provide a logical progression through Barton's journey.
2) The tone: Barton's writing is sharing the knowledge he has gained from experts. Both the framing of his own journey and his writing are humble and makes it much more readable (and also more constructive as I didn't feel defensive).
3) It is research-driven: this book emerged from educational research and stays true to that. Barton's emphasis on the research and his highlighting of disagreements in the literature lends great credibility to his writing.
1) The structure: each section is broken into several sections that make it very easy to follow. I also really appreciate the comprehensive references and the summary sections at end of each section. The chapters (each made up of several sections) also provide a logical progression through Barton's journey.
2) The tone: Barton's writing is sharing the knowledge he has gained from experts. Both the framing of his own journey and his writing are humble and makes it much more readable (and also more constructive as I didn't feel defensive).
3) It is research-driven: this book emerged from educational research and stays true to that. Barton's emphasis on the research and his highlighting of disagreements in the literature lends great credibility to his writing.
April 17, 2020
A stunning book which completely changed my views on teaching Maths. I could completely sympathise with the narrative of this book being a journey from what Mr Barton "thought" worked well anecdotally, to what "actually" works well according to research and how this can look in, practically, the secondary maths classroom.
A must read for any secondary maths teacher, and all the more powerful if you already have a few years under your belt getting by on what you think works (is this the hypercorrection effect in action?!).
A must read for any secondary maths teacher, and all the more powerful if you already have a few years under your belt getting by on what you think works (is this the hypercorrection effect in action?!).
June 27, 2018
I bought this book after attending a professional development session led by Craig Barton earlier in the year. Despite having been a Maths teacher for more than 20 years I found this book informative and very interesting. Craig took his findings on cognitive science and summarized them in this book in a practical way that teachers can immediately go away and implement. I would highly recommend this book to any maths teacher, regardless of how experienced they are.
April 29, 2020
This book is amazing. The problem is that I now need to spend a lot of time working out how to apply this to my own teaching and in turn to convince the rest of my maths department to make some important changes consistently.
November 24, 2019
A amazingly inspiring book for a math teacher!! While reading I've already started changing my teaching methods like ''silent teacher'', problem pairs and diagnostic questions!
August 14, 2025
My students will thank me for all of the torture strategies I have found in this book 📚
December 13, 2020
I went gangbusters on this book at the beginning, as I was pretty frustrated with how my students were learning. This book gave me some great new ideas based on research. As I got pretty far into it, though, I found myself not sleeping as well (I usually read right before bed) because I kept rehashing what I was going to do the next day. So then I started avoiding it a little bit, and it took me a while to get to the end.
Overall, great book. Each section gives an explanation of what the author used to think (common misconception about teaching), several references/research articles for background and additional reading (which I felt were rather pointless because I wasn't going to go look at them when I had the summary right in front of me), and then an explanation of how the research has affected his teaching.
My big takeaways are daily quizzes to help students keep their understanding, diagnostic quizzes to make sure students know the prerequisite material for each lesson, and purposeful practice that takes small steps from problem to problem in order to help students see patterns and think mathematically.
In the end, I wasn't really clear on how it all fits together in a lesson, and how pacing works in his classroom. The book is written from a UK perspective, so some of the references to standardized tests and curriculum were lost on me. But overall I was encouraged by the many practical and sensible ideas presented. Sometimes I wondered about the value of the ideas since it seems a lot of the ideas are new to author (having developed them in the last year or two as of its writing), and some of the ideas did kind of seem like common sense. But with all the confusion present in education about what is good and what is bad, it's probably necessary.
So I got his follow-up book to read, "Reflect, Expect, Check, Explain," which shows promise of explaining more of the details of how it all works.
Overall, great book. Each section gives an explanation of what the author used to think (common misconception about teaching), several references/research articles for background and additional reading (which I felt were rather pointless because I wasn't going to go look at them when I had the summary right in front of me), and then an explanation of how the research has affected his teaching.
My big takeaways are daily quizzes to help students keep their understanding, diagnostic quizzes to make sure students know the prerequisite material for each lesson, and purposeful practice that takes small steps from problem to problem in order to help students see patterns and think mathematically.
In the end, I wasn't really clear on how it all fits together in a lesson, and how pacing works in his classroom. The book is written from a UK perspective, so some of the references to standardized tests and curriculum were lost on me. But overall I was encouraged by the many practical and sensible ideas presented. Sometimes I wondered about the value of the ideas since it seems a lot of the ideas are new to author (having developed them in the last year or two as of its writing), and some of the ideas did kind of seem like common sense. But with all the confusion present in education about what is good and what is bad, it's probably necessary.
So I got his follow-up book to read, "Reflect, Expect, Check, Explain," which shows promise of explaining more of the details of how it all works.
November 29, 2021
Really interesting! I'm having a mini-book club w/ Izzy's math teacher about this.
Some sections I found especially compelling:
2.8 Achievement and motivation
Posits "achievement leads to motivation" instead of vice versa
I've been saying I'd rather Izzy be in a strong math class getting B's than an easy math class getting A's but it did make me think, one shouldn't underestimate the importance of kids feeling success as they go
6. Making the most of Worked Examples
In this book he argues walking through prepared examples w/ no discussion is the most efficient initial way to teach (contrasted w/ socratic or investigatory methods). With Asha and Izzy now when I go through a problem I am trying that would where I work carefully and slowly thru an example w/ no discussion to start... obviously it's a simplification but this whole section was interesting to me. Along with 3. Explicit instruction
7. Choice of Examples and Practice Questions
Highlights the importance of GOOD counter-examples... that's one where it would take a lot of work but I see how it would help. Section 7 "Choice of Examples and Practice Questions" was very interesting to me! I'm the type of math student who benefits from seeing a lot of worked out examples-- as opposed to just abstract proofs-- and definitely like seeing counterexamples , boundary examples etc.
The whole subject of math education is really interesting!
Some sections I found especially compelling:
2.8 Achievement and motivation
Posits "achievement leads to motivation" instead of vice versa
I've been saying I'd rather Izzy be in a strong math class getting B's than an easy math class getting A's but it did make me think, one shouldn't underestimate the importance of kids feeling success as they go
6. Making the most of Worked Examples
In this book he argues walking through prepared examples w/ no discussion is the most efficient initial way to teach (contrasted w/ socratic or investigatory methods). With Asha and Izzy now when I go through a problem I am trying that would where I work carefully and slowly thru an example w/ no discussion to start... obviously it's a simplification but this whole section was interesting to me. Along with 3. Explicit instruction
7. Choice of Examples and Practice Questions
Highlights the importance of GOOD counter-examples... that's one where it would take a lot of work but I see how it would help. Section 7 "Choice of Examples and Practice Questions" was very interesting to me! I'm the type of math student who benefits from seeing a lot of worked out examples-- as opposed to just abstract proofs-- and definitely like seeing counterexamples , boundary examples etc.
The whole subject of math education is really interesting!
April 9, 2021
Rich in insights and great ideas that will enhance any teacher’s practice. A few examples:
- The difference between how novices and experts think, and how that blinds teachers (experts) to the type of misconceptions that students (novices) tend to develop.
- The importance of success to creating motivation, and how lecturing about a “growth mindset” can go wrong.
- How to carefully choose precisely scaffolded problems in order to isolate various points of difficulty.
- The perils (stress, frustration, cognitive overload) of throwing learners into complex problem-solving situations before they have the domain-specific knowledge to be successful.
- How to work with, rather than against, the limitations of students’ working memories.
However, I would be cautious about adopting Barton’s approach wholesale— the supporting research (and his real-world experience) is much stronger in some cases than others. Towards the end, he is suggesting tactics that he admits have never been researched in a math-learning context, or that he had only been using for a year or less.
Also, be aware that the book is mostly about high school math(s), so if you’re working with younger students, not all of the methodology is directly applicable, particularly that which relies on students to engage in metacognitive analysis of their own learning.
- The difference between how novices and experts think, and how that blinds teachers (experts) to the type of misconceptions that students (novices) tend to develop.
- The importance of success to creating motivation, and how lecturing about a “growth mindset” can go wrong.
- How to carefully choose precisely scaffolded problems in order to isolate various points of difficulty.
- The perils (stress, frustration, cognitive overload) of throwing learners into complex problem-solving situations before they have the domain-specific knowledge to be successful.
- How to work with, rather than against, the limitations of students’ working memories.
However, I would be cautious about adopting Barton’s approach wholesale— the supporting research (and his real-world experience) is much stronger in some cases than others. Towards the end, he is suggesting tactics that he admits have never been researched in a math-learning context, or that he had only been using for a year or less.
Also, be aware that the book is mostly about high school math(s), so if you’re working with younger students, not all of the methodology is directly applicable, particularly that which relies on students to engage in metacognitive analysis of their own learning.
January 29, 2023
Essential reading for high school math teachers, not because it presents some sort of cookbook on how to become an effective teacher, but because it honestly chronicles the author's evolution as an educator. Teachers of all stripes would do well to follow Craig Barton's example: Stay current on the research, reflect on your practice, and —above all— keep putting forth your best effort every day.
December 29, 2023
This is a substantial work, based on a lot of academic research as well as the author's practical experience.
It's contains a lot of detail and is therefore quite hard going, but it's written in a simple and clear manner.
If what it proposes delivers the sort of improvements in student learning it suggests are likely, it's a book every maths department should be seriously considering.
It's contains a lot of detail and is therefore quite hard going, but it's written in a simple and clear manner.
If what it proposes delivers the sort of improvements in student learning it suggests are likely, it's a book every maths department should be seriously considering.
August 16, 2021
I am always dubious of education research or rather its applicability in practice, and so found some of the claims of the book questionable. But there are so many good points and fine takeaways that even a skeptic has to find value in this book.
June 2, 2024
This is a phenomenal book absolutely jam packed with useful tips on teaching, rooted it cognitive psychology. I also appreciate the humility and honesty with which it was written. This feels like a book I will continue to come back for myself and for teaching my son over the next decade plus!
July 27, 2025
As a Science teacher with a background in Biology who found herself teaching Maths, this book was fundamental to build my teaching algorhythm and feel confident I wasn't letting my students down. I even translated some ideas over to my Natural Sciences lessons. Many thanks!
July 28, 2019
Highly recommended read for any 6-12 math teachers out there!
August 24, 2019
Definitely had conflict with this author but in a great way. Most useful book in teaching math(s) thAt I’ve ever read. So detailed and well supported and eminently useful. Well worth a read.
October 6, 2019
A lot of great, really useful advice. A few chapters I skipped due to being aimed more at secondary maths teachers.
July 19, 2020
It definitely made me think.
October 20, 2020
Great resource
So much good examples of how to apply evidence based teaching techniques in the classroom. If your a math teacher get this book.
So much good examples of how to apply evidence based teaching techniques in the classroom. If your a math teacher get this book.
July 24, 2024
Probably the best work related book I've read
December 29, 2024
It changed my whole perspective on education and now I'm in a bit of a maths edu craze. I recommend anyone who does any form of maths teaching to read this.
May 14, 2025
A great read - probably could have been edited down a bit, but so many insights and practical applications from research and experience.
December 28, 2019
Excellent. Just needs an index and a bibliography to be complete.
August 30, 2020
Yep, I loved it.
Practical, concrete, research-based models for teaching secondary math with clear examples? What's not to love? It's exactly the kind of book I wanted that basically proclaims: "Yes, there's a way to teach math well through teacher-led instruction." Students can communicate with each other. Students can problem-solve. Students can "think critically." It's such a sigh of relief. Elevating direct instruction IS possible!
A lot of education textbooks will say something like, "Formative assessment is a necessary part of your instruction. Here are ten types of formative assessment... thumbs up/down... clickers... exit ticket..." Cool, we know.
Craig comes at you like, "Hey! Here's everything I was doing wrong with formative assessment and what actually matters. Here's this crazy specific process and procedure I do that has yielded this result and here's 10 worked examples of the types of questions I ask."
For example, he has "SSDD problems" which stands for "Same surface, different depth" problems. Four questions surrounding the same diagram in the center, each corner a different follow-up question pulling from algebra or geometry. It trains students to find connections between different mathematical topics and asks them to pivot their understanding accordingly. Lovely idea because a.) you can look up his website and see 100s of these organized by topic and b.) you can now write your own and grow your own ideas from it.
Craig's ideas aren't "far out" or even original. I'm sure SSDD problems don't sound too impressive right now. It's more that Craig has structured his instruction around 10-12 routines that serve a specific function. "Diagnostic questions" (10-second multiple-choice questions at the beginning of class) are his answer to an effective way to see where students are before instruction. His highly specific method for "low-stakes quizzes" are his answer to teaching students that the skill of recalling material in a test-like setting is necessary for long-term retention. He's carefully constructed these routines and taught his students the intentional practices.
Think of it like Late Night TV. The format is predictable: monologue, interview, game, interview, game, musical performance. It's entertaining. You learn something about the celebrities. I imagine that students look forward to learning in his classroom because of these clear routines. He pulls out an SSDD problem and they know exactly what they're in for.
I have thoroughly enjoyed Craig's ideas behind some of the curriculum questions that haunt me the most at night. I love that it is written to the skeptic. Now I'm a convert. Yes, the book is a little repetitive at times. Yes, there's a little bit over-emphasis on efficiency. But I wish I could shove this book into my hands during my sophomore year of college. Then again, I don't think I was quite ready then.
* * *
Here's a little glimpse of the book:
What's wrong with the following multiple-choice question?
"Which of the following is a multiple of 6?
a.) 20
b.) 62
c.) 24
d.) 26"
Answer: Consider that it isn't truly assessing whether students know the difference between a factor and a multiple. A better question would have had 2 or 3 as an option. (Oh, by the way, the answer is C.)
Yeah, his book is that specific (and it even gets better after this). Anyway, if you enjoy thinking about how high schoolers learn math, you're in for a real treat.
Practical, concrete, research-based models for teaching secondary math with clear examples? What's not to love? It's exactly the kind of book I wanted that basically proclaims: "Yes, there's a way to teach math well through teacher-led instruction." Students can communicate with each other. Students can problem-solve. Students can "think critically." It's such a sigh of relief. Elevating direct instruction IS possible!
A lot of education textbooks will say something like, "Formative assessment is a necessary part of your instruction. Here are ten types of formative assessment... thumbs up/down... clickers... exit ticket..." Cool, we know.
Craig comes at you like, "Hey! Here's everything I was doing wrong with formative assessment and what actually matters. Here's this crazy specific process and procedure I do that has yielded this result and here's 10 worked examples of the types of questions I ask."
For example, he has "SSDD problems" which stands for "Same surface, different depth" problems. Four questions surrounding the same diagram in the center, each corner a different follow-up question pulling from algebra or geometry. It trains students to find connections between different mathematical topics and asks them to pivot their understanding accordingly. Lovely idea because a.) you can look up his website and see 100s of these organized by topic and b.) you can now write your own and grow your own ideas from it.
Craig's ideas aren't "far out" or even original. I'm sure SSDD problems don't sound too impressive right now. It's more that Craig has structured his instruction around 10-12 routines that serve a specific function. "Diagnostic questions" (10-second multiple-choice questions at the beginning of class) are his answer to an effective way to see where students are before instruction. His highly specific method for "low-stakes quizzes" are his answer to teaching students that the skill of recalling material in a test-like setting is necessary for long-term retention. He's carefully constructed these routines and taught his students the intentional practices.
Think of it like Late Night TV. The format is predictable: monologue, interview, game, interview, game, musical performance. It's entertaining. You learn something about the celebrities. I imagine that students look forward to learning in his classroom because of these clear routines. He pulls out an SSDD problem and they know exactly what they're in for.
I have thoroughly enjoyed Craig's ideas behind some of the curriculum questions that haunt me the most at night. I love that it is written to the skeptic. Now I'm a convert. Yes, the book is a little repetitive at times. Yes, there's a little bit over-emphasis on efficiency. But I wish I could shove this book into my hands during my sophomore year of college. Then again, I don't think I was quite ready then.
* * *
Here's a little glimpse of the book:
What's wrong with the following multiple-choice question?
"Which of the following is a multiple of 6?
a.) 20
b.) 62
c.) 24
d.) 26"
Answer: Consider that it isn't truly assessing whether students know the difference between a factor and a multiple. A better question would have had 2 or 3 as an option. (Oh, by the way, the answer is C.)
Yeah, his book is that specific (and it even gets better after this). Anyway, if you enjoy thinking about how high schoolers learn math, you're in for a real treat.
December 20, 2020
I enjoyed this book very much! I could identify with each and everyone of the author’s “misconceptions” about maths teaching! I can’t wait to apply what I have learned when the next school year is starting!!
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