Beyond Pizza and Pies Book Study Part 3

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Welcome to week 3 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 can't believe we are over half way done this book!  I feel like it is a fairly quick read yet I have some great information to take back to the classroom with me in the fall.

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 5: Is 1/2 Always Greater Than 1/3? 

"Our research suggests that many students did not understand that, within a context, the size of the wholes must be determined before comparing fractional quantities." 

This quote from chapter 5 really sums up what I have seen over and over again with students and teachers. When I learned about fractions in elementary school, I can't recall a single time where we solved a problem about fractions where the wholes were not the same.  Even now, when companies send me various curriculum samples, many of them do not contain any problems in which the wholes can vary.  I think a lot of the issue with kids not understanding that the size of the wholes must be determined is that they don't get a chance to see a lot of problems where the size of the whole might vary.   

If your curriculum is lacking in this area, the activities suggested in this chapter will go a long ways toward helping your students uncover this important fraction concept  The pattern block activity described in 5.2 and in the video clips is one I have done with students many times and it is a great way to bring forward a variety of big ideas around fractions  I also love the ideas presented in 5.2 around using 6, 12, and 24 packs of soda or water.  This is definitely somethings I will be thinking about adding to my repertoire for next year.  

In my school, we have a long standing tradition of introducing the ideas of the size of the wholes with Hershey bars.  A good teacher friend of mine came up with this many years ago when we were co-teaching a fifth grade.  Now I do this lesson in grade four.  I get two paper bags and some candy bars.  One bag contains regular sized Hershey bars (the ones with 12 small rectangles).  The other bag contains the smaller Hershey bars.  Here you can either use the minis or the small ones with 4 rectangles that often come in an 8 pack at drug stores and supermarkets.  I then hold one bag and have another person hold the other bag.  Then I ask, "Who would like a chocolate bar?  Mrs. ____ and I have chocolate bars in these bags.  I will be giving out whole chocolate bars and Mrs.  _____ will be giving out a half of a chocolate bar.  Please raise your hand if you want a whole chocolate bar and I will come give you one."  Most or all of the kids choose the whole chocolate bar which of course means they get one of the small ones.  If no students choose the half, I will "run out" of whole bars and one kid will have to settle for half of one.  When it is reveled that the kids who is getting half actually gets more, many kids are shocked but I think it leaves a big impression about making sure you notice the size of the whole.  

Does your curriculum include lessons about the size of the whole or ask questions where the two wholes vary in size?  

Chapter 6: How Come 1/5 is Not Equal to .15

I was shocked by the statistics presented in the "What's the Research" section of this chapter.  It really scared me that so many students had no concept of how fractions and decimals are related.  The fact that almost half of the sixth graders in the study couldn't write 4.5 as a fraction really made me question the way fractions and decimals are taught.  In most cases in a given year, I cover fractions first and then kind of glide into decimals after.  I try to make sure I am showing how they are related and do many activities that involve fractions and decimals during our decimal unit but I am thinking about how I can make the work with fractions and decimals even more explicit.  It also brings up the question about which topic to teach first and how much of one topic (fractions or decimals) I should teach before I start using both simultaneously,  These are good questions to ponder as I begin planning for next year.

In previous years, we had probability units in grades 4 and 5 where a lot of work was done with fractions and decimals between 0 and 1.  Now that we have moved to using the Common Core standards, the probability work has shifted toward grade 7 and we no longer do these units in grades 4 and 5.  I want to make sure my students are still getting some of these opportunities to work with fractions and decimals together.  I like how many of the activities suggested in this chapter are game based.  I am going to make sure we have a set of math stations next year that really work on this fraction and decimal connection.

Two activities I added a few years ago when we stopped doing the probability unit that I think can be really helpful and are very similar to the double number line lesson presented in this chapter are the Meter Stick Number Line Lesson and the 100 Bead String Number Line Lesson.  

How do you make sure your students have a good understanding of how fractions and decimals relate?  

Looking forward to reading your responses in the comments below!

Beyond Pizza and Pies Book Study Part 2

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Welcome to week 2 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.  I know many folks are participating by following along.  Even if you are not caught up on the reading, please feel free to comment and share your own experiences with teaching fractions.

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 3: Understanding Equivalency

I love the opening scenario in this chapter.  Kids following a procedure that their teacher taught them without any real understanding of the concept or context is so common in our current mathematics instruction.  If this teacher hadn't dug a little deeper and asked a context question like this one, they might never have known that their students had no understanding of the idea of equivalency.  I have had this happen to me and seen it happen to other teachers over and over again.  When you move from teaching procedures to teaching conceptually, it can be hard seeing how little your students understand.  

I am blown away by the video clips in this chapter and the entire lesson on measuring with Cuisenaire Rods.  I spend a lot of time on equivalency using area models and a number line but this lesson seems like a great way to bridge the two.  This lesson is simple and easy to replicate but seems like a brilliant way to look at equivalency and to move from an area model to a linear or number line model.  The follow up lesson where the rods are used on the number line also looks very effective.  These two lessons will definitely be a part of my students' work on equivalency next year.  

Fraction Kits: Friend or Foe?

I love the subtitle of this chapter.  I have a love/hate relationship with fraction kits.  When I first started transitioning to using more manipulatives and visual models, I thought fraction kits were the best thing ever.  Then I realized many of my students were using their fraction kits like calculators.  Something that was meant to be used as a tool to move them toward conceptual understanding was now being used as the only strategy they had for solving problems.  Since then I have been on a quest to find the right balance between hands on experiences and helping kids to develop mental models and conceptual understanding.  

I found the research the author conducted about using fraction kits to be similar to my own experience.  I love when this happens!  She talks about how she worked with 2 different groups of students on part-whole fraction ideas.  Cuisenaire Rods were used with both groups.  One group used them as a fraction kit where each piece was always the same fraction.  The other group used them with more flexibility where the whole would change and therefore the names of the other pieces.  The second group made more gains in their understanding than the first group.  When I started using pattern blocks in addition to fraction kits, I saw the same thing.  Although problems were initially harder for kids as the whole was changing, the developed a better understanding of fractions over time. 

Based on this reading and my own experiences, I think fraction kits still have a place and I will be keeping my making fraction strips lesson in third grade.  However, I will be adding the lessons in the book using Cuisenaire Rods and will continue to think about how to make sure students are not using fraction kits as calculators.  

I leave you with this quote from the end of the chapter:

"They (fraction kits) can support students' reasoning about fractions and help them make sense of basic fraction computation.  When used in a superficial way, however, fraction kits may lead students to develop superficial understandings of part-whole relations.  Students may come to understand fraction names such as one-fourth to be merely the name of a piece from a fraction kit, not a name that implies a specific mathematical relationship between a part and a whole." 

What are your experiences with fraction kits? How about teaching equivalency? Let us know in the comments below!  

Join us next week as we look at the importance of context in identifying the unit and making sense of fraction and decimal notation.  Be the first to see new posts by following my blog (look in the upper right hand hand column for ways to follow me) or my Facebook page.

Beyond Pizzas & Pies Book Study Part 1

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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!