Exponents & Logarithms the MTBoS way

A new semester meant a fresh start to try some new things in my maths classroom that would hopefully turn the dial of cognitive demand to appropriately awesome.

Here’s what I did with my clean slate:

I read The Classroom Chef

These guys (Matt Vaudrey & John Stevens) hit the nail on the head repetitively without missing the nail at all throughout their book, The Classroom Chef.

For me, this book nearly gave me whiplash from nodding my head so often and with such enthusiasm. It was so inspiring to read about two other teachers’ experiences in the maths classroom and their journey to deliver the best lessons that they could to their students. Additionally, I have been able to connect with many other teachers around the world who have also read the book and feel the same way. Matt and John are both very active on Twitter (@MrVaudrey & @Jstevens009) and their book has added so many incredibly talented maths teachers as members to the MathsTwitterBloggoSphere (look out for the hashtag: #MTBoS). Here are a few highlights that captivated me as the reader:

This book really helped me gain some perspective about my own pedagogy and how I was dedicating time in lessons. A notion that I still ruminate on when planning for a lesson is giving up on fast food teaching. By avoiding the quick and easy way of teaching and opting for the more nutritious option and trying to plate it up in an appetising way, I am able to serve up better learning opportunities to my students. Apologies about the cooking terms!

Matt and John also created a website for their book which adds an amazing (digital) page to their book: classroomchef.com/links. Here, they have collated lists of resources created by a range of teachers active in the MTBoS. This is one of the best collections I have seen and the blogs, resources, and other websites this has led me to continue to add value to my lessons. It easily trumps my post, Five Resources I wish I knew about when I started Teaching Maths.

I wanted students to develop the NEED to learn “maths”

The first topic that I was teaching my new classes was Exponents & Logarithms. When I was a student, I was able to master the algebraic procedures to rearrange and manipulate exponential equations to get the correct answer without much of a conceptual understanding at all. Something that I had only learnt recently, through teaching, was the reason why certain “laws” or “rules” work and the underpinning theories behind them. I realised that equality was a nice theme to help understand these kind of manipulations and that there are multiple ways to represent the same thing or number.

On day 1, I started with this activity (something that I have now run with groups of teachers too). This was inspired by the Classroom Chef and Mathematical Mindsets (Jo Boaler – @joboaler):

Pick any number.
This can be your favourite number, lucky number, or whatever you want it to be.

Write down the number in as many ways as you can.
For example, if you chose the number “9”, you may represent it as 3 x 3, 3^2, 8+1, or any other way that equals 9.

Choose your favourite representation and copy it onto the board.
Here, I encouraged students to copy the representation that they thought was most creative.

Draw your number on the board.
I asked the students to first think about how they could draw their number, this could have been a picture, shape, or any other visual way of representing their number.

Here are some photos of what they came up with:

I needed to ask better questions

Over the previous semester, I had been utilising Mary Bourassa’s (@MaryBourassaWhich One Doesn’t Belong model to frame questions and problems. What I found really useful about these types of problems was that ALL students could have an answer. I asked all students to commit to one answer, which can sometimes be challenging in itself, but what I was really interested in was the reasons why they picked the answer that they did. Recently, I have spent a lot more time talking with my students about their gut instincts and the perception they have with various things in mathematics. Through hearing about their classmates’ thoughts and choices, and comparing these to their own, the students have begun learning more about their own mathematical intuition. They are starting to notice different things that they may not have noticed or thought about prior to hearing another person’s opinion.

Here is how I used the WODB format to help my students explore equality and understand multiple representations (for more, see the WODB page or click here):

I quickly found that students were noticing many different things about each representation (some that I didn’t anticipate or notice myself!). I had so much success in how the students engaged with these, so I started asking questions like:

  • What do you notice about each one?
  • Which one(s) definitely belong?
  • What might the fifth box look like?
  • Do any of them belong?
  • When “?” appears, what belongs in the “?” space?

We used these problems to explore the different ways we can use index laws to represent a number and solve problems. This worked quite well but, quite quickly, I found myself removing more and more information. Even the question marks! I think I was inspired by this nice way of teaching logarithms. Check out what I mean below:

Wait a second… I’ve seen these before!

Surely I wasn’t the first person to take information out and replace it with a blank space like this. It turns out I definitely was not. So, I jumped back onto the links page of Classroom Chef and re-discovered Open Middle. After bookmarking the page, I searched the website for deliciously challenging problems for my students. Here’s what (and who) I found:

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The amazing people responsible for writing these awesome problems were Erick Lee (@TheErickLee), Nanette Johnson (@Math_m_Addicts), Zack Miller (@zmill415), and Bryan Anderson (@Anderson02B). I loved these so much, but couldn’t help falling into old habits by making Open Middle style problems using the WODB format.

When Open Middle met WODB…

Pick a positive integer and make each box equal that number.
By “each box” I made it clear to the class that I was talking about the four quarters of the square. You’d be surprised about how confusing that instruction can sound!


I asked the following questions to help students describe the thought process they went through when completing each box:

  • Which one did you find easiest to make equal to your number?
  • Which one did you find most challenging?
  • What would be an easy number to start with?
  • Were there any boxes that could have more than one correct answer?

Then, I gave them this one:

Make each box equal to the same number.
In this problem, students should notice that the top-left box has no blanks and equals 8.


From 50 students, only 1 student found the correct answer to the bottom-right box without the help of Google. 49 students got stuck and developed a need to learn how to solve a logarithmic equation. Success.

I’ve put out a call for help to get ideas about how to develop the need to learn the rest of the log laws, it turns out that the MTBoS is a great place to start. Click on the tweet below to check out the response to my call for help below:


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