Friday, February 24, 2017

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!


  1. I might have to try that math challenge with some of my kids, but first I'd have to figure it out! :)

    1. Grab some cards and play around! It is tons of fun and super engaging! It took me a while to figure it out the first time I did it but I wasn't keeping track of my attempts which meant I was starting from scratch each time. Writing down the order you had them in goes a long way towards getting the problem solved.

  2. I love this book! I am planning to start one of my upcoming lessons with a problem first. I also want to do more number talks/number strings with visuals. At first, I may show the visuals as they talk, as a model, then let them do the talking as well as show the visuals. I also want to ensure that students are talking more than I do--convincing skeptics of their answers.

    1. Excellent point about making sure the students are doing more talking than you! I think it would be fun to have someone keep track for me when I am teaching what percent of the time I am talking versus the kids are. That might make me way more aware of what is happening right in front of me.