Mathematical Mindsets Chapter 7

From Tracking to Growth Mindset Grouping

Welcome to Week 7 of our Mathematical Mindsets book study! Chapter 7 addresses a very traditional approach to higher level mathematics, and provides significant research against it. The topic is Ability Grouping. Have you heard of it? Have you experienced it, either as a student or as a teacher? I most certainly have, and Dr. Boaler’s research lines up so well with things that I’ve experienced with ability grouping, and she’s opened my eyes to effects I hadn’t really thought of before.

Dr. Boaler, along with an arsenal of research, argues that ability grouping is detrimental to a growth mindset. Grouping students at an early age into low, medium, and high achieving math classes not only hurts the self-esteem of the students in the “low” group; it also denies them the opportunity to ever achieve certain high-level math courses, since their “remedial timeline” is already mapped out and only reaches so far by the end of high school.

Ability grouping also creates a fixed mindset in the students who are in the “high” groups. If they believe they are naturally good at math, when they eventually come to a point where they struggle, they will be less likely to persevere.

Teaching ALL students to have a growth mindset towards math is key. Dr. Boaler shares several strategies and examples of teaching math to heterogeneous groups in a way that fosters growth in all students and allows students to take multiple pathways of achievement. Below, I share some that I am planning to implement this school year.


Open-ended math tasks

When working with students at a variety of levels, open-ended math tasks are key. These “low floor, high ceiling” tasks, as Dr. Boaler describes them, allow students to build a solid foundation and take the skill to as complex a level as they are willing and able to. I needed some more input on what exactly classifies a quality, open-ended math task, so thankfully Dr. Boaler pointed out a few places to look!

The first is part of the UK’s Nationl STEM Centre, and is a collection of task cards that are designed for mixed-achievement groups. These cards are part of a project called Secondary Mathematics Individualised Learning Experinece, or SMILE! Some of the wording will need to be changed, as they were made for students in London, but I have already found some great resources to use during our first unit on Decimals. I definitely need to remember to change things beforehand, though, or my kids will look at the word form of 0.7 and go “What the heck is “nought point seven,’ Miss G.?!”
I encourage you to visit this website and check out the extensive library of SMILE cards and other STEM resources. It’s free to sign up for login access to download the materials.

Another resource that has yet again proved useful is the YouCubed math website. Below is a link to a perfect example of a “low floor, high ceiling” task, which allows kids to explore different ways of approaching the problem, and allows them to go as in-depth as they are able.




This Ice Cream Scoop problem is also the perfect way to introduce multidimensional values, which Dr. Boaler shares as another key aspect of heterogeneous teaching. Allow for, value, celebrate, and assess multiple ways of thinking about and approaching mathematics! 

Don’t focus solely on performing calculations, but encourage students to “ask good questions, propose ideas, connect different methods, use many different representations, and reason through different pathways” (Boaler 121). And remind students regularly that “No one is good at all of these ways of working, but everyone is good at some of them.” This mindset will set the tone for group work that shares responsibility and is respectful of everyone’s thinking.


Roles and Shared Responsibility in Group Work

Dr. Boaler recommends spending a significant amount of time and effort at the start of the year teaching students how to effectively and appropriately work in groups. She shares many strategies for this, but the ones that I am ready to jump right into are:
  • ·         Work with students to carefully develop group norms of respect and listening

o   Create posters of what students do and don’t like other group members doing when working together. This provides student awareness and ownership over norms that we would give anyhow, such as not letting one person do all the work and tell everyone else the answer, saying things like “this is easy” when it may be confusing to some, leaving people out of discussions, etc.
  • ·         Assign roles for students in groups that allow them to balance the workload and equitably participate. Roles such as Facilitator, Recorder/Reporter, Resource Manager, and Team Captain can categorize specific jobs that must be done and questions that can be asked to stem discussion.
  • ·         When assessing group work, assess only one randomly selected group member’s verbal or written response. If they cannot clearly articulate the way their group found the answer, the group must reconvene and continue discussing until every group member clearly understands the concept. They can let you know that their ready to be reassessed, and you ask the same student to explain. Check out Tried and True Teaching Tools' awesome Role Cards created from the roles Dr. Boaler shares - Kathie did an AMAZING job on these, and I will definitely be using them! 
I am so grateful for this book study, and the way in which this book is opening my approach to teaching mathematics! I hope you will join our conversation here in our blog hop comments, or in our Facebook group at Mathematical Mindsets Book Study Facebook Group{request to join, we’d love to have you!}

If you’re interested in some of the research cited in this chapter, I’ve linked some of Dr. Boaler’s studies below.

Mathematical Mindsets Chapter 6

Mathematics and the Path to Equity

Thanks for joining us for Week 6 in our Mathematical Mindsets book study! This chapter was jam packed with strategies, so I want to jump right in!

The statistics that this chapter opens with are alarming. As we continue to learn with this book, there is no such thing as a good-at-math gene, so I figure, “Why focus on the past?” Women and minority groups have statistically been left in the dust and are underrepresented in higher-level mathematics, so how do we, as teachers today, prevent this from continuing to happen?

I’ve decided to focus this post on Dr. Boaler’s Equitable Strategies, which she shares as “strategies for purposefully making math more inclusive.”


Offer all students high-level content

We need to make sure all students have the opportunity to attain higher-level mathematics courses. Dr. Boaler addresses this in greater detail in chapter 7, so I will hold off on discussing in detail.


Work to change ideas about who can achieve in mathematics

The mindset beliefs held by teachers open or close the pathways for students, and that fixed mindset thinking and teaching is a large part of the reason inequities continue in math and science, for women and students of color (Boaler, p. 102).

Give students the message that you know they can succeed in math. And don’t just say it, actually know in your heart that every single one of your students has the potential to succeed!


Encourage all students to think deeply about mathematics

Girls have a greater tendency than boys to want to understand deeply why methods work, where they come from, and how they relate to other concepts and domains (Boaler 2002b; Zohar & Sela, 2003). When we focus solely on the procedural aspects of mathematics, we are denying the opportunity for deeper, meaningful understanding. The following aspects of successful math teaching can guide us in solving this problem:

Hands-on Experiences – Providing direct interaction with the workings of a concept can greatly increase a students’ conceptual understanding, as well as their ability to form connections to other concepts and disciplines. I began implementing Math Centers in my upper grade classroom last year, and directly saw the benefit of hands-on experiences. Here is are examples of a Fractions center, where students created posters with a variety of models for the concept of dividing a whole number by a fraction:

Project-based curriculum – This year, we finished out our year with Digital Divide and Conquer’s Final Frontier outer space PBL unit, which phenomenally connected many of our math concepts in a project-based, engaging format. Matt has created TONS of great PBL resources, and I highly encourage you to check out his shop:

Curriculum with real-life applications – I try to tie mathematics into our engineering challenges and find other ways to make real-life connections, but this is definitely an area I need to focus my growth on!

Opportunities to work together – A study of Berkeley students in high-level math classes highlighted the importance of working together in mathematics. High-achieving Chinese-American students were observed completing assignments in a collaborative manner, supporting each other’s struggles and working through challenges together. The African-American students were observed completing assignments in isolation, and were quick to give up when struggles arose, because they felt they were just not good at the math. This led to alarmingly high failure rates, but was completely turned around after researchers provided seminars on collaboratively approaching mathematics. The African-American population actually surpassed the Chinese-American population within 2 years of the seminar’s implementation, proving the importance of collaboration and a positive mindset!

I plan on encouraging more collaboration in my math block by continuing centers and really incorporating math talks using the MP’s as often as possible. Angela Watson has a fantastic set of cards with number talk question stems to help us build math discussions, which I’ve linked here:


Eliminate (or at least change the nature of) homework!

I know the topic of homework can lead to heated discussion, because it is so customary to U.S. schooling. There is so much research out there, however, that homework has no impact or a negative impact on student learning. If something is not helping out students’ learning, we should reconsider it and make some changes. If something has been proven to have a negative impact, we should throw it out the window like it’s on fire!

If you want to make the jump into eliminating math homework, here are some resources Dr. Boaler lists as evidence in support of this:
  • Alfie Kohn – The Case Against Homework
  • Sal Khan – The One World School House
  • Various resources from Challenges Success, 2012

If you’re not quite ready to ditch homework altogether, or if your school requires that you assign homework, then Dr. Boaler recommends at least changing the nature of that homework: “Instead of giving questions students need to answer in a performance orientation, give reflection questions that encourage students to think back on the mathematics of the lesson and focus on the big ideas” (page 108). 

I’ve adapted Dr. Boaler’s example of this into the sheet below, which you can download as a freebie to get you started on this homework shift!

Whew! I know this was a long post, but the strategies felt so valuable to me that I felt compelled to inspire other teachers to use them. Which of the above are you willing to give a try? Comment below, and hop through the link up to see other bloggers big takeaways!

Mathematical Mindsets Chapter 5

Mathematical Mindsets Chapter 5

Wowzers…I feel like the last few weeks have been a time warp of wedding planning and summer school, and I have been so bad about keeping my posts a priority, but I must say, all the wedding planning is exciting!! I have a growth mindset that I can improve my blogging though, and I’m determined to stay on track with this book study.
I thoroughly enjoyed Chapters 3 and 4, and I hope you check out some of the other bloggers thoughts on key takeaways. Chapter 3 shared the importance of teaching the beauty and creativity in mathematics, and Chapter 4 shared some fantastic games and activities for building number sense.

Creating Rich Mathematical Tasks

My biggest takeaway from Chapter 5 was that it is in our hands to ensure we are providing our students with rich, engaging mathematical tasks. Dr. Boaler shared 6 cases of mathematics instruction that hooked the learners by piquing their interest and presenting a challenge that they were determined to solve, almost as though the math problems were brain teasers!

My favorite of these was the number talk on 18 x 5 that Dr. Boaler conducted at a staff meeting for a modern, tech-savvy online course startup company, Udacity. As people shared their methods for finding the product, Dr. Boaler drew out the visuals of their thinking on the whiteboard table. She shares that the buzz of excitement in that room, and in any of our math classes when experiencing a task like this, is that “most people…have never realized numbers can be so open and number problems can be solved in so many ways.”
So how do we incorporate this magic and wonder into our elementary and secondary classrooms?!

3 Tips to implement:


Ask the question before teaching the method. (pg. 81)
Dr. Boaler acknowledges that most of us are provided a curriculum from which to teach. She spends so much time in this chapter, however, giving examples of how we can modify the WAY in which we introduce the concepts so that kids have a brain teaser/math challenge approach to spark curiosity. Dr. Boaler suggests giving the students a challenging problem that incorporates the concept in a real, meaningful way, without first telling the students the process to solve it. Give them time to try things, brainstorm, and discuss first. Doing so will make the introductions of formulas and algebraic properties so much more meaningful for the kids once we do introduce them.


Open math tasks to encourage multiple methods, pathways, and representations. (pg. 77)
Find ways to open up math problems to multiple avenues of success, such as the 18 x 5 number talk. I can’t wait to revisit my math curriculum and see where I can do this, and I am definitely thinking of starting the year with number sense activities like this. I think these are PERFECT ways to introduce and begin using the CCSS Math Practices.


Add visual components.
Visualizing our thinking as well as the thinking of others, and discussing the models, is so valuable to our understanding of complex mathematics concepts. The more we can provide this for our kids, the better!

Thanks for sticking with me! I hope you can apply some of these strategies to your teaching to help make math more exciting and meaningful for our kids! Don’t forget to check out the other thoughts in the link up below, and please comment and share how you might incorporate these strategies! I’d be particularly interested in any 5th grade/upper elementary insights!!