#FractionsFebruary Blog Tour

I am finishing out the #FractionsFebruary blog tour put on by Math Solutions.  I had the privilege of interviewing Julie McNamara, author of Beyond Pizzas & Pies and Beyond Invert & Multiply.  I love reading about teaching fractions and always feel like there is more I can learn.  I read both of these books about a year and a half ago and they brought great changes to my teaching practice, especially the idea of using cuisenaire rods in my fraction lessons.  Beyond Pizzas and Pies is great for grades 3-5 and Beyond Invert and Multiply is great for grades 4-6.  If you teach 4th or 5th grade, I highly recommend you read both of these books! 

What is the most exciting piece of research on teaching fractions that has come out over the last 10 years?

Siegler, Thompson, and Schneider’s (2011) work highlighting the importance of students’

understanding of fraction magnitude provides much needed insight into the importance of the

number line. They found that students who had a good understanding of fraction magnitude (as

evidenced by their ability to accurately place fractions on a number line and to accurately compare

two fractions) were also more successful with problems involving fraction computation. Siegler,

Thompson, and Schneider suggest that the extension of students’ “mental number line” to include

rational numbers is an essential aspect of numerical development - what I refer to as number sense

and fraction sense.

What do you feel is the biggest misconception students have about fractions? 

This is a tough one but one big misconception is that fractions are always considered in terms of

food (pizzas, pies, brownies, etc.) and that fractions are not numbers.

What can teachers do to help students overcome this misconception?

One thing teachers can do is to build on students’ early work on partitioning areas, with explicit

attention to the importance of equal partitioning, and connect this to linear measurement models.

Contexts are very beneficial, especially ones that can be considered on a number line like time and

distance. One of the reasons I use Cuisenaire Rods so much in my work is that they are concrete

enough for students to manipulate and they can be used as bridges to work with number lines.

What was the most surprising thing you learned in the process of writing your books?

I found that I really liked the process and that I LOVE thinking about fraction division.

If you were to go back into the classroom as a teacher, what grade would you choose to teach and why?

This is another tough one. I really love fourth grade, as there are so many opportunities to help

students begin to think abstractly. I also love middle school because there are so many connections

back to the mathematics of elementary school that students are often hesitant to consider. They

don’t know how much they know! One of the best parts of my position at the university is that I have

many opportunities to go into classrooms and work with students in local schools.

You can read the rest of Julie's interview over at the Math Solutions Blog!

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Creating Mathematical Mindsets

Welcome to week 2 of our Mathematical Mindsets book study! Last week, we looked at the latest in brain research and talked about the importance of mistakes and struggle.  

Chapter 3: The Creativity and Beauty in Mathematics

Big Ideas

- Math gets treated differently than other subjects.  It is much more of a performance subject than any other.

-There is a big gap between real world mathematics and school mathematics

- Students (and the public in general) see math as calculations, rules and procedures rather than creative and beautiful.  

Impact in the Classroom

Fibonacci Numbers and the Golden Ratio

One of the examples given in this chapter was about taking a look at the Fibonacci numbers and the golden ratio with kids.  I have always loved this series of numbers and many years ago was given a copy of Fascinating Fibonaccis which I have now used many times to talk with kids about Fibonacci numbers.  I also love the picture book Rabbits Rabbits Everywhere by Ann McCallum who also wrote Eat your Math Homework.  One more excellent resource is this blog, Fabulous Fibonacci Fun where you can find more background information as well as a great collection of images that illustrate the golden ratio in nature. 

Let Students pose Their own Questions 

Real mathematicians pose and answer questions all the time.  One of the best ways to get our students doing this is to let them have the chance.  Give them mathematical tools or situations and let them come up with questions.  Let them work to find answers to their questions.  This is something I have had some opportunity to do this year in the context of our school wide genius hour.  It is definitely something I hope to do more of in the future.

Chapter 4: Creating Mathematical Mindsets: The Importance of Flexibility with Numbers

Big Ideas

-Kids intuitive joy of math is quickly replaced with learning procedures & rules

-Students need to see math as a conceptual, growth oriented subject.  They should see math as a place to think, not to blindly operate on numbers.

Impact in the Classroom

Number Talks

The single best way I have found to develop a sense of numeracy in kids of all ages is number talks.  Doing number talks with my students has been a total game changer in my students' ability to think about numbers, develop strategies and learn to talk about their thinking.  It is a structured way to spend 10 minutes each day that will give you big results.  If you teach K-4, start with this book, and if you teach grade 5 and up check out this one.  If you have been using number talks in your classroom and are ready to up your game, there is a new number talks book all about Fractions & Decimals. I have dug into this one over the past 2 months and it has really helped move my practice forward and increased my students' understanding of fractions and decimals. 

Hold off on Formal Procedures

When do you "teach" kids the traditional algorithms for adding, subtracting, multiplying and dividing?  Holding off on these formal procedures can really help kids see math as a growth oriented subject where they can develop their own conceptual understanding and design their own strategies to solve problems.  If you are using number talks, you will be amazed at what kids of efficient strategies your students can come up with on their own.  In my school, we have agreed not to introduce the traditional algorithm for addition and subtraction until grade 4.  We hold off on the traditional algorithm for multiplication until the end of grade 5 and division until grade 6.

Go for Depth, Not Speed

Value deep thinking over fast thinking in your classroom. Whenever I talk about not focusing on speed, the issue of math facts comes up.  I think to much emphasis is put on memorization when kids should really be working on knowing facts from memory.  For much more about facts, check out this post. 

Chapter 5: Rich Mathematical Tasks

Big Ideas

-5 c's of engagement: curiosity, connection making, challenge, creativity, collaboration

-5 ways for teachers to open math tasks and increase potential for learning.  

    1. Open the task so there are multiple methods, pathways and representations

    2. Include inquiry opportunities

    3. Ask the problem before teaching the method

    4. Add a visual component and ask students how they see the math

    5. Ask students to convince and reason; be skeptical

Impact in the Classroom

Rich mathematical tasks are such a good way to generate engagement and enthusiasm in your classroom.  Providing kids with low floor, high ceiling tasks are a great way to get big math ideas, work on perseverance and get kids excited about math. After reading this chapter, I decided to try a little experiment with a very engaging, very open ended task, the 1 to 10 card investigation. If you have never heard of this investigation, head here to check out a 1 minute video. 

I presented the original investigation to a group of first graders who are always looking for a challenge, all of my second graders, the entire 5th and 6th grade classes and a large group of teachers during a PD day. All of these folks were able to access this problem.  The excitement and engagement were just as high with adults solving this problem as it was with first graders.  

Students work together using 20 frame playing cards to solve the problem to 15. 

The best part of this problem was that there really is no end to it.  After folks solved the 1 to 10 card problem, all kinds of extensions were proposed and worked on.  Some kids increased the number of cards while others proposed different arrangements of the cards such as what if we flip one over and then put 2 on the bottom.  Kids challenged each other and me to go further with this problem.  One of my sixth graders must have spent 20 hours on this problem over the last week and was able to generalize a pattern that would work for any number of cards.  Kids asked to take cards home and challenge their families.  Teachers who worked on this problem during PD literally could not stop working on it. 

A student uses cards from my place value to 120 deck to work on solving the problem with 50 cards

 Your turn!  What did you think about this week's reading? What changes are you making or thinking of making in your classroom?  Please respond in the comments below!

The Power of Mistakes and Struggle

Welcome to week 1 of our book study of Mathematical Mindsets. 

Chapter 1: The Brain and Mathematics Learning

Big Ideas

-Folks who have a fixed mindset can develop a growth mindset.  Their learning approach can become much more positive and successful.

-The students who show the highest achievement in mathematics around the world have a growth mindset.  This can put them a full year ahead of other students. 

-Folks with a growth mindset are more likely to do something perceived as hard. They see mistakes as a way to grow their brain and motivation to keep going.

-Fixed mindsets can develop as a result of praise given by parents and teachers

Impact in the Classroom

Teach kids about brain research.  

Research about the brain and growth mindset has come a long way over the last few years.  I have found Jo Boaler's Week of Insiprational Math to be a fantastic (and free!) way to show kids about how the brain works and get them excited about math.  We started our school year with the week of inspirational math in grades 3-6 and have recently re-watched some of the videos included in the lessons to refresh ourselves on brain research.  We have plans to do a week of inspirational math in our K-2 classes in the near future.  

     Another excellent resource for helping your students understand brain research is a series of short cartoon videos produced by Class Dojo. Recently, we started k/1 math each day with one of these videos which led to some great conversations about how the brain works.  Even if you don't use Class Dojo, these videos are a great resource!  It looks like you can check out the first one on YouTube and the rest are available on their site.  It is free to use and only takes a minute to sign up.

 Change praise

 Praise from teachers can help kids develop a fixed or a growth mindset, it all comes down to the type of praise they receive. Praise has such a strong impact, it can affect their behavior right away.  The praise kids receive in the classroom needs to switch from  you are so smart to you are working really hard. This has been a challenging switch for me because despite good intentions, it is so easy to fall back on the old reliable praise.  Like most things, it is something teachers need to practice.  We spent some time at a staff meeting brainstorming different things that fit into the category of praise and then classifying it as praise that would develop a growth or a fixed mindset.  This is an area where it is so important for kids to get a consistent message and it is definitely worth discussing as an entire staff.  A colleague also shared this Growth Mindset poster with us and posted it in several prominent teacher areas. It has been a helpful reminder and has continued the discussion about changing our praise.

Educate Families

If students are really going to change their mindset than both teachers and parents need to be aware of this research and have practical strategies to help them.  Since our entire school has been working on kids developing a growth mindset, we have communicated with families in several ways.  We started the year by showing a few of the videos from the Week of Inspirational Math at our back to school open house.  This was after we had used these videos with kids so many families had already heard parts and pieces of this.  We also use our school wide newsletter to communicate what we are doing and share growth mindset tips.  The topic has also come up during parent teacher conferences, especially with those kids who need the most work on their mindset.  We have also shared this great little parent handout with families.   It is still a work in progress to educate families about growth mindset but we have made a start and are now developing a common language for kids at school and at home.  

Chapter 2: The Power of Mistakes and Struggle 

Big Ideas

- Mistakes grow student's brains

- When a student makes a mistake, there is increased electrical activity in their brain.  

- Even if you don't know that you made a mistake, it still grows your brain

- After a mistake, brain activity is greater for those who have a growth mindset rather than a fixed one. 

- Successful people are the ones who make more mistakes than non-successful people

Impact in the Classroom

Give students the chance to make mistakes

Giving students a chance to make mistakes is one of the best things you can do to develop their understanding of mathematics.  If they make no mistakes day after day, then the math you are giving them is not challenging enough.  Kids come to believe that being smart means getting the answer right on the first try and with a bit of speed.  I love this video from the Teaching Channel where Carol Dweck author of Mindset: The New Psychology of Success talks about personalized learning and how important it is for kids to make mistakes. 

Teach students that mistakes are positive

I am a big fan of this routine called My Favorite No.  I do a lot of formative assessment in my classroom and doing My Favorite No is a great way to directly address misconceptions with the class.

A few weeks ago, I found these growth mindset posters and put them up on a bulletin board.  In my fifth grade class, I have many kids struggling with growth mindset so I decided to really use these posters with them.  I decided to have them place a sticker on the poster each time they used the "think" bubble.  It struck us very quickly that we make mistakes all the time.  The mistake poster became full of stickers while the others just had a few.  This has been a great way to show how common mistakes are and how important they are to learning.

Test and Grade Less

Our school uses standards based grading for our report cards which means I test and grade a lot less than I used to.  However, there still are times when folks want percent grades, particularly for Special Education Evaluations.  We also have added several "screeners" and other forms of testing as we have moved toward an MTSS model in our district.  I do a lot of formative assessments so I have a really good handle on where kids are.  I am on board with less grading and testing but I am not sure my school/district is ready to change much more in that direction right now.  I am excited that chapter 8 is all about assessment and hope to be able to try some new ideas.

Your turn!  In the comments below, let us know what you thought of these 2 chapters.  If you have a blog and want to post there, just share the link in the comments below. Haven't read the chapters?  Share your ideas about mistakes, struggle, brain research or anything else related to growth mindset!  I look forward to some great discussions! 

Math Love, A Giveaway, A Sale, Book Study Update & More

First Grade Number Puzzles

Book Study Update

Our new book study is getting a great response!  We will be discussing Mathematical Mindsets by Jo Boaler.  This book has the potential to make a great impact in  your classroom regardless of what grade you teach.  It is also super affordable and ships for free if you have Amazon Prime. Get all the book study details here.  

A Sale

Also in the news this morning, Teachers Pay Teachers site wide sale starts tomorrow!  You can save 28% off all of my resources with the code LoveTPT.  I have been window shopping already and currently have 18 items in my cart.  There is nothing like a good sale!

I have been so inspired by my K/1 math group over the last month and have been super busy creating resources to use with them.  I have quite the range of abilities in this group and routinely need to create stuff that spans the K-2 spectrum for these guys.  I am loving teaching math in a multi-age setting and will have more to share about how we structure this very soon.

Math Love

Lots of new Valentine/heart themed math centers for K-2 classrooms!

Counting Puzzle Bundle

Valentine Ten Frames Playing Cards & Activity Set

Teen Number Math Craft

Addition Facts Craft

Kindergarten Number Puzzles

Second Grade Number Puzzles

Math Love Giveaway! 

Thanks for all the love you show me all year long!  Enter below for your chance to win a $10 TPT gift card!  I will send it to the winner before the end of the TPT site wide sale on Wednesday so they can use it during the sale if they wish.

  a Rafflecopter giveaway

Growth Mindset and Mathematics

What does a student's past experience have to do with how they learn math today?  My past experiences certainly shaped me and changed my mindset several times.

My  Math Story

I was told from a young age that I was gifted in math.  I learned math easily and was always ahead of my peers.  I can not ever remember struggling with a math problem until I was in fifth grade.  When I was a fifth grader, my teacher allowed me and another student to work on our own, starting at the back of the textbook and working our way to the front.  As a teacher, this sounds like a terrible idea in retrospect but it did lead me to develop some perseverance that had seriously been lacking. After that, I continued into middle school and high school continuing to do well in math.  I had a great memory and was really good at remembering procedures, formulas and rules.  Now I know I was getting by on this and not on a deep conceptual understanding of math.  

I knew I wanted to teach elementary school from the time I was in tenth grade.  I went to college as an elementary education major.  We had to pick a second major and I randomly picked sociology.  Luckily, someone convinced me that having a background in math would be much more marketable so I switched my second major to math.  This led me to taking 2 math classes per semester.  When I got a class on non euclidean geometry, I hit a major wall.  The content was challenging and required so much more of my focus and attention then I was used to.  I worked harder during that one class than I had on all the other math classes I had taken combined.  It gave me the experience of working hard and still barely keeping my head above water.  I don't remember much of the content of that class now, but I do remember what it was like to struggle and to work really hard to learn math.  

Despite a major in math in college, I still lacked the conceptual understanding of math that I needed to teach.  I started teaching sixth grade and really struggled with how to help my students who didn't have a good foundation in mathematics.  I spent a lot of time repeating the steps of a procedure louder and slower until my students could do it.  After a few years of teaching sixth grade, my position was cut and I moved onto a new school in a new role as a math specialist.  I got the job because I could do a lot of math, not because I knew a lot about teaching it.  I was given a lot of free rein to use my role anyway I wanted.  I spent a lot of time the first year just figuring things out.  I started listening to my students more and helping them develop strategies rather than teaching them.

From there, I delved into some of the deeper research available on teaching math at that time.  I read a lot of books and got involved in some really good professional development.  I took more math courses, these ones aimed at teachers and at developing a deep understanding of content.  I did a lot of group work and spent a lot of time learning from my peers.  I saw multiple approaches to solving problems and expanded my definition of what it meant to be good at math.  For the first time, I saw the connections between algebra and geometry and gained a valuable understanding of big conceptual ideas.  This conceptual understanding of mathematics led me to be a much better teacher.

Mathematical Mindsets

Nowadays, I am so much better at helping my students learn math.  I have so many more tools, strategies, ideas and a much better understanding than when I first started out.  Despite all this, I still have students who are struggling.  I know that I have the tools I need to help them but something big seems to be getting in their way.  Their mindset.  The research on the brain and mindsets has been moving quickly over the last few years and now their is so much available information to help students, teachers and parents with mindset.  My research on mindset last summer led me to Carol Dweck's Mindset book.  From there, I found Jo Boaler's Book Mathematical Mindsets and knew it was the next thing I wanted to work on.  I got the book in July and read it almost in one sitting. It got my wheels turning as I was starting to think about back to school season and I have made some big changes this school year based on ideas in this book.  Now that I have the basics down, I am reading this book again (a little slower this time!) and am going to go deeper into some of the ideas as well as share how I have used some of it in my classroom.  Over the next 9 weeks, I am going to take one chapter at a time and take a closer look at these ideas.  I would love for you to join me!

Book Study

I read a lot of math teaching books!  From time to time, I like to dive deeper into a new book or an old favorite by hosting a book study on my blog. I have done book such as Minds on Mathematics, Beyond Invert and Multiply, Mathematics Through Play, Number Talks, Children's Mathematics,Whole Brain Teaching For Challenging Kids, Teach Like a Pirate and Guided Math in Action. It is fun and easy to participate.  I will post my thoughts and share some ways I am using the ideas in my classroom each week and then you can share in the comments section.  If you have your own blog, feel free to post there and leave the link in the comments section. 

This book is easy to access and very inexpensive for a good professional development book.  You can grab it on Amazon, it ships free with Prime.

Here is the posting schedule
Friday February 17th: Chapters 1&2
Friday February 24th: Chapters 3-5
Friday March 3rd: Chapters 6&7
Friday March 10th: Chapters 8&9

What is your math story? How did your mindset change as you learned math? Please share in the comments below!