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Which Comes First... Decimals or Fractions?

Let me start by saying I have been meaning to write this blog post for a long time... a very long time!  It was around November of this past school year that I had a revelation (thanks to a fantastic graduate course and professor, the amazing research-based book Extending Children's Mathematics: Fractions and Decimals, and students who were so willing and able to communicate their math understanding and struggles).  The order in which I have been teaching the 5th grade math standards was not allowing students to make critical connections between fractions and decimals.  Here me out... There are certain concepts in math that are pretty obvious in which order they should be taught.  Clearly students should learn to add before they learn to multiply, or divide whole numbers before learning to divide decimals.  What I found is not so clear among many teachers and those district leaders who write our pacing guides is what comes first?  Fractions or decimals?  Both of these concepts play a huge role in 5th grade math and provide an imperative foundation for math learning through high school and beyond.  Don’t believe me?  Hear what Sherry Parrish and Ann Dominick, authors of Number Talks: Fractions, Decimals, and Percentages have to say about the lack of strong fraction understanding in high school students. 

“The National Mathematics Advisory Panel conclude that the most important foundational skill not presently developed appears to be proficiency with fractions... The panel’s findings were corroborated with a survey of 1,000 U.S. algebra teachers, who indicated that a lack of fraction knowledge was the second biggest problem students faced in being prepared to learn algebra” (2016, pg. 2).  

If this is one of the biggest problems students face in learning algebra in high school, it is important that we do all that we can to teach fractions (and decimals and percentages) in a way that allows our students to truly and deeply understand fractions.  Now that you know how important mastery of these concepts is, which should you plan to teach first?  The short answer: fractions.  Then why do many pacing guides do the exact opposite?  Honestly, I have no clue.  I have asked the creators of several different pacing guides for their rationale for teaching all of the Number & Base Ten standards first and then teaching all of the Fraction standards next and nobody could explain to me the reasoning behind this.  Truth moment… For a long time I have taught it this way because that is what the pacing guides have always said to do and I thought surely this was the best way.  It was not until I did my own learning and challenged what I had always done did I realize that decimal understanding is strongest when it builds on students’ understanding of fractions.  See if any of this sounds familiar:
  • Why is 0.01 (one-hundredth) smaller than 0.1 (one-tenth)? Why is 0.2 one-tenth of 2?  To explain this, you need to build on students' fraction understanding of what “one-tenth” truly is. 
  • Where do I put the decimal when I multiply two decimal numbers together?  No matter how many models we used, this question never seemed to disappear UNTIL I got around to teaching multiplication with fractions.  At that point, they could explain why 0.1 x 0.01 = 0.001 or 1/10 x 1/100 = 1/1000.
  • What does it even mean to divide 5 by 0.2 and why do I get a bigger answer?  Why do I get a smaller answer when I multiply 5 by 0.2?  I thought multiplication was supposed to make numbers bigger and division was supposed to make numbers smaller! Yes, technically you can explain this without necessarily getting into fractions, but clarifying this by building on students’ understanding of fractions is much less abstract than attempting to explain this by building on students’ struggling understanding of decimals.  In addition, the models used to teach multiplying and dividing fractions answer these two questions in a visual way perfectly.
Check out the learning progression for decimals and notice how embedded many of the fraction standards are in the learning of operations with decimals in 5th grade!  Believe it or not, decimals are much more abstract than fractions.  The authors of the book Extending Children's Mathematics: Fractions and Decimals state that "children's understanding of decimals simultaneously draws on their understanding of fractions and place value" (2011, pg. xxiv).  I will say, sometimes students find "temporary success" more quickly with operations with decimals than they do operations with fractions if the instruction is focused on procedures rather than conceptual understanding. Procedures for operations with decimals are typically more familiar to students than the procedures for operations with fractions.  But, is this really true understanding? Does this type of learning (procedural rather than conceptual) last and allow them to make critical connections in their growing understanding of fractions, decimals, and percents?

Instead of teaching decimals and constantly jumping ahead into my fraction unit to explain why or how, I finally decided to just take a break from our decimal unit, move right into our fraction unit, and let students discover all the why's and how's for themselves!   By the time we came back to our decimal unit, I could direct students to think about their own understanding of fractions to answer their many questions about decimals. (Side note: You could teach operations with fractions and decimals in conjunction with each other because they have so many connections.)  At the end of my fraction and decimal units, I spent a week or two reviewing and taking time to intentionally highlight the connections many students had already made about fractions and decimals throughout the units. Luckily I have the flexibility to deviate from the pacing guide, so I am excited to use what I have learned from watching and listening to my students' thinking, paired with some great research, to come up with a new plan!

What are your thoughts?  Which comes first on your pacing guide, fractions or decimals?  In what order do you choose to teach fractions and decimals?


*This post contains affiliate links to two great math resources to assist with the maintenance of this blog!

Fractions: Classroom to Real World

Have you ever had a student ask... "When am I ever going to use this?"  Early in my teaching career this question would drive me crazy!  It usually was asked by a student who was capable of doing whatever we were learning, but simply didn't feel like actually doing the work.  This question now makes me realize I have not fully been doing my job as a math teacher...  There should never be a time when a student does not understand the purpose or application behind what they are learning!  When students understand when they would actually use the content they are learning, their motivation is immediately boosted.  Not only do students become more motivated in learning the content, but they are also challenged to apply (Bloom's Taxonomy, anyone?) the content which typically requires more critical thinking.  This is why there has been such a big push for "real world connections" in recent years.  As math teachers, we must take our math instruction from classroom use to real world use and fractions is the perfect place to start!

After visiting the Ron Clark Academy a few years ago, I was totally inspired to use classroom makeovers as a way to engage my students even further in our real world math projects!  As our fraction unit comes to a close, I wanted to share some pictures of all the fun!

We hosted a cooking show to share our delicious dessert recipe with the world! Students got to "bake" and apply their skills adding and subtracting fractions to some real life cooking situations!  I simply threw some plastic table cloths on the tables, brought in cooking utensils, and played some bakery music and we were good to go!  Even though the floor was covered in powdered sugar at the end of the day, it was worth it!


The math cafe was under construction while we used our skills multiplying fractions to build model dog houses!  I added caution tape to every place I could think of around the classroom, put out some safety cones, and added all sorts of construction-like centerpieces to the tables!  I even found an hour long video on YouTube of a construction site (what non-teacher would ever use this?!).  I threw on a pair of safety googles and a reflective safety vest to really get into the construction spirit! :)


Despite the cold weather, we ran our own lemonade stands in the classroom to help us practice dividing fractions!  I reused the plastic table cloths, let students design lemonade stand signs, and brought in yummy treats for them to have at their lemonade stand!  I played some Jack Johnson in the background as kids worked because it just made me think of warm weather (I was impressed that several students actually knew who Jack Johnson was!).  This was the easiest classroom makeover and the kids had lots of fun!


We still have our fraction review project, which is a picnic theme, that we will do some time this week, and then it's off to our next unit!  I don't makeover my classroom for every project we do, because honestly my students are engaged with the real world scenarios of the projects anyways!  But, I do believe the classroom makeovers bring an element of surprise and joy to my students' faces that is just too addicting to resist! :)

First Week of School Activities

Can you believe it is back to school time already?  We just finished up an amazing first week of school and I could not have been more exhausted!  With a summer full of fun and relaxation, my body had forgotten how much energy I use teaching all day!  After a weekend of rest, I am looking forward to the second week of school.  But first, I had to share with you a few of the activities I did with my students in their first week of the new school year!

Play Dough Introductions
Before the kiddos arrived, I placed a little container of play dough on each of the students’ desk.  The Dollar Tree has 4 mini-containers of play dough for $1, so it’s not expensive at all to get a class set and they come in a variety of colors!  Since I don’t do a seating chart this early in the year, I gave students a little card that had a color on it and asked for them to sit at a desk that had play dough the same color as their card.  This takes the stress out of finding a seat for any nervous students!  Once they were at their seat, they were directed to take out the play dough and make anything they wanted to as long as the object they made had the same first letter as their name.  While they were making their creations, I was able to move around to every table and just spend time with my new students, make them laugh, ask them questions, and just put them all at ease.  After awhile, each student shared their name and what they made.  Later in the day, if I couldn’t remember their name, I would simply ask them what they made and I was usually able to come up with their name!  By the end of the second day of school I had all 48 of my students' names down!  All I had to remember was Emma Elephant, Nate North Carolina, Trey Tree and I was good to go! J  Students had fun, we all got to laugh, and it took pressure off me and the students!

A Twist on Teaching Procedures
Let me start by saying this, I hate teaching rules and procedures.  It’s not fun for me and it’s not fun for the kids, but it is a necessary foundation I have to build in order for the rest of the school year to be successful.  After being inspired by Hope King at Elementary Shenanigans, I decided to get my students out of their seat and have a little fun so I didn't bore my students to tears with procedures.  I started the activity by explaining to students that I had a few procedures we as a class need to commit to following.  I told them I was going to give them a quick run down of them, but they had to pay very close attention to them because I had a little game where only those who remembered this information would be successful.  It’s all about selling it... the kids were hooked and they listened very closely so that they would be prepared for whatever task I threw at them later.  After I ran through the procedures, I paired up all the students and gave each pair two stretchy headbands (another Dollar Tree find) and a set of cards that had the different rules and procedures on them.  Students took turns putting the card in their headband while their partner acted out the procedure.  The student with the card in their headband had to guess which procedure their partner was acting out.  It may sound cheesy, but the students had a blast and I laughed so hard my cheeks hurt!  After going through the deck a few times, I brought all the students back together and we played a little game of “What Would You Do?”  I gave the students different scenarios that might come up in the classroom and then asked the students “what would you do?”  Students answered the questions using the correct procedure they had just acted out.  Even though I highly dislike teaching rules and procedures, this turned out to be so much fun!


Recipe for Success
Even more important than rules and procedures are the expectations we have for how students contribute to the culture of the classroom.  Character traits such as respect and integrity must be present from all students if the classroom is going to be a place where students feel safe to make mistakes, share their ideas, and participate in learning.  Intentional math talk is one of my focuses for this school year, so I realize it is more important than ever to establish these expectations early!  Instead of simply telling my students these expectations, I decided to do a baking activity to represent our Recipe for Success in the classroom.  I made these delicious Chocolate Chip Cookie Dough Balls in front of my students as we talked about what each "ingredient" meant.  We discussed how all of these ingredients by themselves are important, but when they are all mixed together they create something to be desired by all students.  When we as a class uphold all of these expectations (respect, effort, integrity, teamwork, responsibility, and positivity), our classroom becomes a place that everybody loves to be a part of!  You can download the freebie I made to go along with this baking activity below!  The recipe makes enough for about 30 students and is super easy since it doesn't require any baking.  Be careful though, these balls of yumminess will ruin any diet you plan to start with the new school year!  


Now that we have laid the groundwork for how our classroom will run, we will spend the second week of the school year building a solid math foundation!  Students need to be taught certain math “behaviors” and grow a mathematical mindset if they are going to grow as mathematicians.  I’ve been looking forward to this week since the end of last school year, so I can’t wait to share with y’all some of my favorite resources for building a strong math foundation! Stay tuned J


Math Talk in the Classroom (Part 2)

If you missed the Math Talk in the Classroom (Part 1), be sure to check it out!

Now that expectations have been set for how students will participate in math talk, it’s time to practice talking about math!  Yes, we must give our students the opportunity to practice talking because they don’t always know how to talk the way we want them to.  In order to practice talking about math, it is important that students are talking about something they are comfortable with so that the focus of the discussion is on the talk and not the content.  Before students can wrestle with rich content, they must first understand how to communicate their thinking in a productive way. 

Why do students need to practice talking about math?
Students need to practice upholding the expectations for math talk, while also gaining experience listening and interacting with the reasoning of others.  In the book Classroom Discussions in Math, the authors touch on the importance of students understanding that they may agree or disagree with an idea, not a person.  When students learn to agree and disagree with an idea, rather than a person, the discussion becomes more focused on the content and students’ desire to learn.  Another way to keep students focused in the discussion is to set the expectation that students must respond directly to or build upon what was previously said.  This minimizes students’ tendency to think only about what they want to say, rather than listening to what others are saying.

Giving students the opportunity to practice talking about math is also beneficial for us as teachers!  It gives us practice facilitating math discussions.  If this is something you are newly incorporating into your teaching routine, this is a great opportunity for you to practice using higher order thinking questions to facilitate the discussion!  

How can we give students practice talking about math?
Now that you know why students (and teachers) need to practice talking about math, let me share with you how I plan to give students this practice.  I am so excited to use the book Which One Doesn’t Belong this year to teach students how to talk math.  The math talk that can happen with this book makes my brain dance!  The book is set up with four different shapes on each page and a question that simply asks students to decide, “which one doesn’t belong”?  All four of the shapes on each page can be the correct answer to the question, which means students will really have to think and talk about WHY each one doesn’t belong.  Because students think differently, they will have different paths of reasoning making for rich discussion (and many opportunities to practice upholding math talk expectations).  Do you see how powerful this book can be?  Below are a couple of images that Talking Math With Kids has provided for you to use in your own classroom!



If you are looking for more resources to use in your classroom to get students comfortable talking about math, Would You Rather Math and the Which One Doesn’t Belong blog are wonderful in that they provide many images that require deep thinking and strong reasoning.

I am truly looking forward to using these resources to give students practice talking about math and to provide students with a fun experience as they grow their reasoning skills.  As I discussed in Math Talk in the Classroom (Part 1), math talk has so many benefits beyond what I have shared in this post.  Check back soon for Math Talk in the Classroom (Part 3), where I will dive into using math talk to further students understanding of mathematical concepts!


*This post contains affiliate links to two great math resources to assist with the maintenance of this blog!

Math Talk in the Classroom (Part 1)


Kids love to talk, did you know that?  Truth be told, I love to listen to them talk.  The things that come out of their little mouths can be downright hilarious.  If you listen close enough, you’ll also notice kids have such a unique (and sometimes surprising) perspective on everything.  So how can we as teachers tap into this to improve math learning in our classroom?  We do this by making math talk a regular and welcomed part of our daily routine. 

What is math talk?
Math talk is simply talking about math!  There is nothing about math that is off limits to talk about!  Talk about math concepts.  Talk about math discoveries.  Talk about math mistakes.  Talk about math vocabulary.  Talk about how students approached a problem or different ways they solved a problem.  Talk about it all!  Math discussions can be had in small groups or as a whole group.  Teachers, we have to be the ones to intentionally facilitate this and teach our kids the importance of talking about math.

If you haven’t read Classroom Discussions in Math, I highly recommend that you pick it up!  After reading it for a graduate course, I was hooked on math talk! The authors (along with many other math gurus) make a strong case for why it is critical and necessary for kids to talk about math in the classroom.


Math talk can help guide our instruction.  There is no better way to learn what our students are thinking than listening to them share their thoughts.  Talking about math reveals what students know and what they do not know.  Math discussions allow students to reason more deeply and remember more clearly.  Math talk can also develop students’ social skills.

In order to have the most productive math discussions, we have to teach our students how to talk math in a way that is beneficial for their learning and the learning of others.  As we head back to school in the fall, one thing I absolutely plan to incorporate into the first weeks of school is teaching students why and how to talk about math. 

Why do we talk about math?
       Talking about math allows us to share our thoughts.
       Talking about math helps us clarify our own thoughts.
       Talking about math helps us understand what we do and do not understand.
       Talking about math prepares us for the future.
Students will be more willing to participate in math discussions if they understand why they are doing it.  When we talk about math, it forces us to organize our own thinking so that we can share our thoughts with others.  When we listen to others talk about math, it gives us the chance to compare our reasoning with the reasoning of others and decide how they fit together (how’s that for Mathematical Practice #3?).  Math talk teaches us how to interact with each other in a positive way, while working towards a common goal.

How do we talk about math?
I plan to come up with our math talk expectations together as a class, but I will be sure to guide students towards several key ideas.
       We will be kind and respectful at all times.
       We will all participate in math discussions. 
       We will listen and respond to the ideas of others.
       We will support one another so that we all can learn.
It is important that a “safe” environment is established so that students feel comfortable sharing their ideas.  Students must understand that listening is just as much for their own benefit as it is for the speaker.  We cannot learn if we do not listen.  We cannot contribute to the conversation if we have not first listened.  Math talk works best when everyone is working together towards learning. 

I hope you are convinced how powerful incorporating math talk into your daily routine can be for your students’ learning! I can’t wait to share with y’all the activities I have planned for the first weeks of school to set the stage for a year of amazing math talk!  If you're ready for some fun, check out Math Talk in the Classroom (Part 2)!


*This post contains affiliate links to a great math resource to assist with the maintenance of this blog!
References:
Chapin, S. H., O'Connor, M. C., Anderson, N. C., & Chapin, S. H. (2013). Classroom Discussions in Math: A teacher's guide for using talk moves to support the common core and more, K-6.

What's to Come?


I am so excited to launch a blog for Mix and Math and share with you the math things that make up my teacher life! So what’s to come on the blog?


Let me start by saying math is my passion! Well, TEACHING math and empowering teachers to teach math is my passion.  Math just has a stigma... It’s boring. It’s too hard. Some people will just never get math. Only nerds like math. Y'all I could go on and on! I’ve heard it all and I don’t want to hear any more! And can I tell you something? Those comments aren’t just from students! Even some teachers feel that way. I truly believe your thoughts on math are based on the experiences you have had in math. So how do we change the “math stigma”? We have to change the math experience we give our students! 

So, I have decided to organize the blog into three sections...

Teach.  "You don’t know what you don’t know."  This is how I feel sometimes about teaching math!  Every year I become a better teacher because I have more and more experiences working with kids as they learn math. Math needs to be understood deeply by students, and that requires high-level instruction.  This section of the blog is a place for me to share what I have learned about teaching specific math content, classroom experiences I have had with students, mistakes I have made, and provide resources that will allow you to feel confident and prepared to teach your kiddos math. 

Engage.  Kids learn better when they are highly engaged in what is being taught.  Let’s be honest, teachers enjoy teaching more when their kids are engaged and excited to learn.  The whole teaching-learning thing is so much more enjoyable when there is creativity, passion, and innovation.  This section of the blog is a place for me to share with you some things I do in my own classroom and things I see in other classrooms to engage students in learning.  Hopefully you will leave these posts inspired to try something new in your own classroom!

Learn.  I am always learning.  I think I love learning as much as I love teaching!  Because of this I love going to conferences, reading books, and collaborating with other educators!  There is always more to learn, and that’s overwhelmingly exciting!  This section of the blog is a place for me to share with you what I am learning about how I can better teach kids to understand and love math!  My hope is that you will learn something new along the way!

I hope you will join me on this journey! Let’s change the "math stigma" together!