Teaching Math With Examples

by Michael Pershan

Rammed full of practical ideas – all of which are beautifully articulated and backed by research – this is a truly wonderful book.
-Craig Barton , Author of How I Wish I'd Taught Maths

I can’t recommend highly enough this very do-able and high-leverage approach to any math teacher or curriculum developer who wants to level up their task design, or just try out something new.
-Kate Nowak , Former high school math teacher and currently Vice President of Product Strategy for Illustrative Mathematics

Genres: Education, Teaching, Mathematics, Nonfiction

126 pages, Paperback

Published January 27, 2021

Reviews

[Patrick · Rating 5 out of 5]

Best math ed book I've read in a while! This book was filled with discussions about practices I've already adopted in my classroom -- and the research to support or discourage them -- and also many new ideas and techniques I'm excited to try. Chapters 5 and 8 were particularly excellent. I really think every high school math educator would benefit from reading this!

[Bowman Dickson · Rating 5 out of 5]

Amazing! Both challenging and affirming, deeply philosophical but wholly practical at the same time. Bravo

[Kerim Ramadan · Rating 3 out of 5]

A quick summary of using examples in class to teach. Even though this is a short book all of the key points could have been summed up in an article and not cost £12.

The ideas themselves are good, nothing particularly ground breaking if you are up to date with current thinking in maths education (in the uk at least). Useful for maths teachers as a tool to reflect on how they do things, although many may say they apply these principles I would suggest that most would do so very superficially and would miss the key features of the good practice described here.

In summary a clear and purposeful description of how examples can be used. I was expecting more discussion on which examples to choose (a topic I don’t see much literature on at the moment) but that could be because I didn’t read the blurb carefully enough.

[Justus · Rating 3 out of 5]

This is a short and direct with a brevity I appreciated. The text itself is only 100 pages and if you ignore the lengthy final chapter -- which is how to extend worked examples to geometric proofs -- the main text is only 80 pages. So you've got a monograph on using worked examples in (primarily) algebra.

It has extremely limited applicability for anyone who isn't a maths teacher -- or perhaps a private maths tutor -- but then it's not trying to serve anyone outside that core audience, so I don't fault it for that.

Why only three stars?

There were a few sections that felt a bit underbaked somehow. His section on feedback is what really crystalized this for me. He talks about how he gives "feedback" by giving worked examples and provides a routine: give the assessment, mark it, focus on just one problem, create a worked example, call for revision.

A question naturally arises: how can you focus on just one problem when a class of students may have many different kinds of errors across their assessment?

The focus on just one problem is often a sticking point when I explain to others how I give feedback. The issue is that quizzes and tests often contain many different errors, often committed by entirely different students. If a teacher gives a ten-question test, there may be five different problems that students need to improve on. How do you choose just one of them?

My own answer to this question is: you can’t. This is an issue with longer exams, and a reason not to give them.

(Keep in mind his "longer exam" had 10 questions.)

Which, okay, I don't necessarily want to see this turn into a book on assessments but I'm not fully grasping how this is supposed to work in practice, especially when we're also told that modern research says you're also supposed to have spaced practice and retrieval practice and interleaved practice that continually assesses students on past topics.

And even within the constraints he gives, I'm still not totally clear on what do when 60% of your class makes Category A Error and 40% of your class makes Category B Error.

Later on he talks about an example with Laura after teaching the Pythagorean theorem. Laura (and by implication, the entire class) got 100% on the missing hypotenuse questions but 0% on the missing leg questions (so 10 question is bad but 2 kinds is okay?) So he created a worked example on missing leg. But what do you do if 8 students also made mistakes on missing hypotenuse?

There were a few situations like that where I feel like just a bit more could have helped flesh this out for me and bump this up to 4-stars.

[Tony Germany · Rating 5 out of 5]

I’ve been teaching maths at secondary school in the UK for 7 years and the feeling I have after reading this book is that of excitement. Not since Craig Barton’s how I wish I’d taught maths have I felt this way, while Craig’s book covers such a broad spectrum of teaching pedagogy, this focuses specifically on how Michael has used worked examples in his classroom. I feel that most of my lessons may see some future benefit from many aspects of this book. He has weaved a collection of pedagogical research into a little book of golden nuggets applicable to any classroom. My bow has another string.

[Eliza · Rating 5 out of 5]

What it’s about: When and how to use examples to teach maths.
My favourite part: The further reading section, because the author just seems really excited to be writing about the people who influenced him and their work, which is nice.
My favourite quote: ‘Congratulations! You just invented “fading out an example”’ (p. 69).

This is the kind of stuff I wanted to learn at uni.
It also casually mentions a lot of ideas from cognitive load theory (“John Sweller’s research is all over this book” - further reading section), but puts it into a more specific context for maths instruction.

[Michael Webster · Rating 4 out of 5]

A good, short read. Provides snippets of research entwined with practical guidance and tips on how to teach better using worked examples.

I use worked examples to teach many parts of the curriculum, however there are some excellent nuggets in there with regards to getting the absolute most of every example. Not necessarily ground breaking but a worthwhile read if you are looking to get the most out of your teaching.

[Pat Nelson · Rating 5 out of 5]

Every education book should be like this, short, to the point, honest about its viewpoint, full of practical advice with a firm theoretical backing. I got a bunch of tweeks to my use of examples from this lovely little book.

[junela melanie · Rating 5 out of 5]

the writing is super clear and digestible. the most important takeaway for me is how studying worked examples isn't necessarily "giving up", as they can be good learning materials, as long as active thinking is happening.

[Pam · Rating 5 out of 5]

Very practical strategies he has tested in his own classroom for years. Shared research to back up the why too!