Blogs I Love Math Literature Freebies About Home

Wednesday, June 10, 2015

Beyond Pizzas & Pies Book Study Part 1

Welcome to week 1 of our book study on Beyond Pizzas and Pies.  For more information this summer's book studies, check out this post.  If you need more convincing about how much this book has to offer, check out this short introduction video!

I am so excited that you are joining me on this journey to learn more about teaching fractions.  My go to book for fraction teaching for the last few years has been A Focus on Fractions.  I learned so much from that book but realized that I haven't read anything on fractions in quite some time so this summer I am making fractions a priority!

The posting schedule for Beyond Pizzas & Pies (all Wednesdays)
June 10th Chapters 1&2
June 17th Chapters 3&4
June 24th Chapters 5&6
July 1st Chapters 7&8

Chapter 1: The Problem With Partitioning: It's Not Just About Counting the Pieces

The title of this chapter really sums up this issue.  Over and over again, kids are exposed to problems that show equally partitioned shapes or number lines and are asked to find the fraction.  With these types of problems being the only ones they see, kids often develop strategies like counting the pieces and they seem to work because the only problems they see have equal parts.  

It can be very uncomfortable to begin to ask kids to solve problems where the parts are not partitioned equally.  Many teachers (myself included!) really balk at this idea at first.  It can feel like you are trying to trick your students.  What you are really doing though is not letting them rely on counting pieces and are helping them think about fractions rather than just repeating a procedure.  Yes, kids who have always seen equally partitioned areas and number lines will have some wrong answers and some disequilibrium when they first work with shapes that are not partitioned equally but this will lead to a better understanding of fractions. 

I love the classroom activity described in this chapter using Cuisenaire Rods to help kids see that fractions are a relationship between the whole and the part.  Our current math curriculum does something very similar to this using pattern blocks.  The idea of the whole changing rapidly during the lesson freaked me out the first year I did it.  I remember having a discussion with the fourth grade classroom teacher about how confusing the lesson would be to the students.  Well I was very wrong.  It was a bit confusing to the adults at first, but the kids loved it.  I could see their fraction understanding deepening and they had such rich discussions about what they were noticing.  To this day it remains one of my favorite lessons.   I love the take on this using Cuisenaire Rods.  I am one of those people who does not use Cuisenaire Rods very often even though I have met countless teachers who swear by their effectiveness.  I can see how this lesson would be a very nice way to deepen my students fraction understanding.  Because we use a similar lesson already using pattern blocks, I think this lesson would be a great for my intervention groups.  

Chapter 2: Top or Bottom: Which One Matters?

The student at the beginning of this chapter who over generalizes the idea that the bigger the denominator, the smaller the fraction could be a story about the way I learned fractions.  One of my clearest school memories was in fourth grade when I remember realizing that this rule worked.  Of course, we had just started our unit on fractions and had only been working with unit fractions.  I plugged right along in my text book thinking I was big business getting ahead of the class and doing the next several pages only to find out later that I really was doing things wrong.   I had no conceptual understanding of fractions and I don't really remember using visual models.  We had a text book and that was it.  My teacher marked most of my answers wrong and "taught me" how to find a common denominator when comparing fractions.  That is the one and only strategy I had for many years.  After having more life experience with fractions, I was able to tell which fraction was bigger or smaller if they were fairly far apart but still had to resort to a common denominator strategy for many fractions pairs.

After several years of teaching fractions the way I was taught, I began to see there was some work I needed to do.  I had found a copy of About Teaching Mathematics in my classroom and began to read it.  I started moving more of my teaching into conceptual understanding rather than procedures.  I was still pretty stuck on fractions and really lacked conceptual understanding myself on this matter.  I took a wonderful class on teaching fractions and read A Focus on Fractions.   This led to a huge shift in my own conceptual understanding which led to a huge shift in my teaching.  Now instead of teaching kids the procedure for finding common denominators, I guide them toward developing a toolbox of strategies for comparing fractions.

One thing I think I still struggle with in my teaching is making sure kids are not over generalizing rules like the larger the denominator the bigger the fraction.  I get so excited when kids make conjectures that I sometimes forget to stop and ask them if it always works.  I recently read a blog post over at Traditionalist Becoming Non Traditional about journal prompts that ask kids if something always, sometimes or never works.  I think adding in journal prompts about these conjectures kids make could really help make sure over generalization does not happen.

Watching the video clips and reading about the activities in this chapter makes me really rethink some of the number line activities I do and change them around to use the Cuisenaire rods.  This seems like a powerful manipulative for number lines and I am currently not using them at all in my fraction instruction.

Now it is your turn!  What did you think about these chapters?  What ideas can you take back to your own classroom?  What are some things you are doing well?  Let us know in the comments section below!


  1. Ok, I can't take it anymore! I ordered the books. Thanks for giving such great recommendations.
    The Research Based Classroom

    1. Hi Brandi! You will love this book! A good understanding of fractions is so important to success in later mathematics!

  2. I love the fact that there's a dvd to see kids in action with the classroom activities! I wish the 12 cm Cuisennaire rods she uses were available commercially.

    1. Cuisennaire rods are definitely something I need to get busy using! I have a great collection of them collecting dust somewhere. I am moving to a real HUGE classroom this summer and they are definitely something I want to put front and center.

  3. I love the way you think about the way you learned (and unlearned!) things as a child. I really believe that kind of introspection leads to a better understanding of our students' thinking, and better teaching!
    Thanks for sharing!
    Linda at Primary Inspiration

  4. This comment has been removed by the author.

  5. I loved reading your post! I remember the days of teaching fifth grade and having students come in knowing how to change improper fractions into mixed numerals and vice versa, yet they had no conceptual understanding of what they were doing or even what an improper fraction and mixed numeral represent. Tools/manipulatives were so important for my fifth graders, and I appreciate how you stress their importance at older grades. I don't know how long it has been since I've even seen a set of Cuissinaire Rods--thanks for sharing how they can be used in understanding fractions. I will make sure to share your book study on Facebook--I am a little late in getting over here to read your first post, but so glad I came...


  6. Thank you for choosing this book. I bought it in March and haven't opened it. Thanks to this book study, it is the first book I am reading of my summer book list.

    I have been teaching math for 22 years and am always looking to improve, so that I can help my kids better understand. I have no difficulty understanding why the kids get so confused when it comes to fractions. The difficulty is finding ways to break their misconceptions. Their vast experience with numbers so far is just whole numbers. It only makes sense that they use the whole number "rules" on fractions. This book is helping me to find more ways to get the kiddos the experience with fractions that they so badly need in order to build understanding.

    I teach 5th grade, but I will begin the year with the 3rd and 4th grade activities from this book in order to build a solid foundation to construct meaning for the 5th grade concepts.

    Thank you again for this book study.