Teaching Math with Examples

I’ve appreciated Michael Pershan’s writing for as long as I can remember. It always gives me something deep to think about, blows me away with its clarity of thought, and is terribly persuasive. I usually walk away from it feeling like I’m missing something, that I’m not seeing the bigger picture. To top it off, Michael’s wit never fails to make me smile. He’s a great guy and friend.

I can’t help but teacher-crush on Michael right now because of how thrilled I am that he wrote a book! It came out earlier this year and is called Teaching Math with Examples.

Despite the fact that I maintain a blog and use Twitter, I don’t enjoy reading online. I much prefer newspapers, physical books, and anything printed. This is why reading Michael’s thoughts in book format is a dream come true for me. It allows me to indulge in his genius without having to stare at a screen in the process.

I must say, the book doesn’t disappoint. Highly readable, practical, and smart, it’s exactly what I expected from Michael. It starts with him making a case for worked examples and detailing a classroom routine that he’s developed that uses them. He goes on to discuss how worked examples can foster problem-solving. Michael then explores how worked examples can be used as a form of feedback after assessments like quizzes and exams. I was lucky enough to learn about this him firsthand at a MƒA workshop he ran a couple years ago. It’s dynamite. He closes with tips on designing worked examples and exploring geometric proof to show that worked examples can be effective in the most abstract mathematical settings.

A big part of what the book does right is how actionable it is. No matter where I was in my reading, I felt I could put the book down and try out what I just read. In fact, I did this several times! This was empowering. The morning after I finished reading about Michael’s classroom routine in chapter 3, for instance, I paired two of his self-explanation prompts with a worked example on trig ratios. It worked well with my students. This happened again a few days later in chapter 5 after Michael put me on to the interesting distinction between “part-task” practice and “whole-task” practice. There he offers up five adaptations of worked examples that are painfully simple to use. With a delicate sprinkling of examples from most levels of K-12 mathematics, I see Teaching Math with Examples appealing to a wide array of educators.

A running theme of the book that I noticed, and one that Micahel highlights, is how worked examples promote access. I appreciate this. By showing a worked example, we invite more students into a discussion about the complex mathematical ideas we teach. I see this each time I use one; no matter where they are in their understanding of a concept, somehow kids get really into discussing math work. Access is a particularly important point this year since a lot of my students have been struggling to bounce back from remote learning. Worked examples have been a pathway into learning for many of them.

In the end, I came to think of the book as a field guide to worked examples. As such, the only thing missing for me is an index. This would make referencing it in the future that much easier. Heck, my cover is already tattered from flipping through it so much. It helps that it’s small enough to fit into my back pocket. As a fierce hoarder of student work, this little book is going to travel with me and come in handy for years to come.

Other than its practicality, another aspect of the book that I like is how Michael marries research to classroom practice. He judiciously cites articles that have impacted how he uses worked examples, but doesn’t do it in a way that feels heavy or too academic. Though he has read a bunch of research and is considerably more informed than many, including me, he never talks above the reader. He presents research swiftly and ties it directly to his experiences teaching math. His approach is lean, focused, and encouraging. Truth is, his seemingly casual references to this article and that article make reading research feel like a thing that all teachers should be doing more of. By the time I got to the last few pages of his wonderfully annotated bibliography, I wanted to print out some articles for myself and dive in.

At one point in the book, Michael gives advice on how to design worked examples. After showcasing one that is perfectly balanced, he says, “Reading this problem feels, to me, like having my hand held by a comforting teacher, leading me right into the middle of a terrific and complex set-up for a fascinating problem.”

He is referring to a specific example, but it resonates because this is how Michael’s entire book felt to me. Chapter by chapter, I feel like he is holding my hand, leading me through the art and science of worked examples. I never feel alone or that he doesn’t see me, the reader. His writing is personal, but tactically reveals a world of worked examples that is much bigger than himself. He welcomes me into that world warmly and instructively. I didn’t find an exhaustive list of worked examples in his book, but Micheal — like any good teacher — knows this wouldn’t be helpful. Instead, he masterfully provides just the right amount of variety — and then lets my hand go. He shows me what’s possible, why it’s possible, and backs it with evidence. In the process, he gives me the confidence and ability to do it on my own. It accomplishes a lot in 126 pages.


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