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Developing Mathematical Habits of Mind = Experienced Problem Solvers

Posted by Cindy Bryant

Sep 3, 2014 6:30:00 AM

Mathematical_Habits_1Have you ever had one of those days when things just weren’t clicking like they should? Then it dawns on you that you failed to carry out one of your daily habits or routines? Most likely you had some distraction, which resulted in you omitting just one simple act. But oh how that one simple omission can wreak havoc on your entire day!!!

Our lives are full of habits – routines, actions, or behaviors that are so well formed that they happen without any thought on our part. There’s just something about our habits and routines that adds organization and structure, helps us to make sense of our world, and overall helps our lives run smoothly. 

Teachers are pros when it comes to modeling good organizational skills and habits. They’re also experts in demonstrating and modeling correct mathematical procedures. But as you and I both know, learning and doing math is more than knowing and understanding how to carry out math procedures.

It’s the mathematical habits of mind, or modes of thought, that enable us to reason about the world from a quantitative and spatial perspective, and to reason about math content (Levasseur & Cuoco, 2009). These habits, a composite of many skills, attitudes, and likings, enable us to behave intelligently when confronted with a problem to which the immediate answer is unknown. They’re the habits that empower us to use our mathematical knowledge and skills to make sense of and solve problems.

There are two types of problems solvers – experienced and inexperienced. It’s been said that, “Inexperienced problem solvers don’t know what to do when they don’t know what to do. Experienced problem solvers do know what to do when they don’t know what to do.”

Experienced problem solvers know what to do when they don’t know what to do because they’ve developed the mathematical habits of mind that enable them to behave intelligently when confronted with a problem. Without being reminded, experienced problem solvers automatically employ the steps in the Polya’s (1945) four-step problem solving process in an attempt to solve a problem presented to them.

1. They begin by first making an attempt to understand the problem and ask clarifying questions that help them interpret and understand the conditions of the problem.

2. They make sense of the mathematical situation at hand and choose a strategy, tools, and/or models they see as relevant and applicable in solving the problem.


3. They use the strategy they’ve chosen and if that strategy doesn’t work, they try a different strategy – all part of perseverance in problem solving.

4. They take one last look at the original problem, the strategy and processes used, and determine if the solution is reasonable and viable.

But the mathematical habits of mind cannot simply be taught or learned by studying a list. They are best learned when students are immersed in classroom experiences that enable them to engage in the learning of mathematics concepts through problem solving, making and using abstractions, and developing and applying mathematical theories. Classrooms steeped in the Common Core Standards for Mathematical Practice (MP) provide ideal opportunities for students to learn and develop these habits as they:


When students have experiences in making sense of and solving problems and communicating and using precise and appropriate mathematics and mathematics language, they have opportunities to develop the overarching habits of a productive mathematical thinker. The productive thinker knows that it may take more than one attempt to find the solution to a problem and it may require trying more than one strategy.

Problems that can be approached from a variety of entry points are always more palatable to students. When you present students relevant, interesting, and challenging problems that can be approached and solved using different strategies, you give them choices. They have choices in to using variety of tools and models for solving the problem. This in turn helps them to reason about, look for, and make use of structure in solving problems. All of this leads to increased opportunities for understanding important math concepts and skills which strengthens their ability to explain their thinking and reasoning as well as critique the reasoning of others. 

You play a vital role in the development of mathematical habits, in conveying a mindset that fosters productive struggle and emphasizing the value that wrong answers can have. You set the stage by embracing a mindset that fosters productive struggle and the importance and value in grappling with problems. The more you engage your students in learning and doing mathematics, the greater the chances of them developing the mathematical habits of mind of a productive mathematical thinker – an experienced problem solver who does know what to do when they don’t know what to do. 

We value your feedback so we invite you to share your questions, comments, and suggestions about teaching and learning mathematics. Thanks!

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5 Websites to Download Long Division Worksheets

Posted by The LearnBop Team

Sep 2, 2014 8:00:00 AM

Long division doesn't come easily to everyone!

But it is a process that students should know and understand.  Division understanding should be developed by using a variety of manipulatives and models so that students grasp the concept of division as either how many in each group or how many groups.  This sets the stage for developing procedural fluency with division. 

While the Common Core State Standards (CCSS) don't advocate a reliance on worksheets, they do stress the value and importance of procedural fluency.  So these worksheets are useful for demonstrating division procedural fluency in carrying out procedures flexibly, accurately, efficiently and appropriately.

Without further ado, below is a list of 5 websites to download long division worksheets for developing and demonstrating procedural fluency with division.



This website provides helpful practice tools for students Pre-K through 5th grade in a variety of subject areas. offers the option to either generate your own long division worksheets, or to download pre-made worksheets for free.

long division worksheets

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This website has over 4,765 math worksheets on a wide range of math topics. Their free long division practice worksheets come in varying degrees of difficulty.

long division worksheets

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This website provides dynamic downloadable math worksheets. It also enables you to build your own. Generate long division worksheets based on specific needs and download them for free in pdf format. 

 long division worksheets

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This website is a great one-stop shop for worksheets on a wide range of math topics. In addition to long division worksheets, provides helpful quick tricks and strategies to help students formulate a stronger conceptual understanding of each topic. 

long division worksheets

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5) The

Here, you can generate custom math worksheets for multiple topics. Their straightforward interface makes it easy to create, download and print long division worksheets. 

long division worksheets

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Topics: Resources

LearnBop & Common Core State Standards (CCSS) Alignment

Posted by The LearnBop Team

Aug 25, 2014 9:00:00 AM

Recently, we got asked a question on our Facebook page:


"My school doesn't follow the Common Core State Standards for Mathematics (CCSSM). Can I use LearnBop?" 


When we're chatting with parents, teachers and math administrators, this question often comes up.

It's a great question, one we've given a lot of thought to as we developed our interactive learning system. Though LearnBop is fully aligned with the Common Core State Standards for Mathematics (CCSSM), adoption of the Common Core is not essential for the use of LearnBop. Our number one priority is closing the math achievement gap by providing a quality program that supports students in learning mathematics. Our program is informed by the CCSSM along with relevant research about problem solving and how students learn best. 

While we do believe that the CCSSM provides key guidelines for quality math instruction and learning, we have a larger mission than Common Core adoption. LearnBop's interactive system, which simulates the one-on-one experience a student might have with a tutor, is designed to meet every single student where he or she is in the learning process.

We've harnessed the expertise of some of the best math teachers in the U.S. to develop our automated tutorials specifically aligned to the Common Core content and practice standards.  The step-by-step guide that simulates working with a one-on-one tutor conveys perseverance in problem solving and productive struggle. The visuals, hints, and videos support students in learning math concepts and skills they are struggling with to become successful problem solvers throughout their lifetimes.

Download our latest guide to get more of our LearnBop's great content for free, written by Presidential Awardee Cindy Bryant!

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About the Community


Cindy Bryant, LearnBop Director of Learning The LearnBop Community is an informal PLC for 5th - 9th grade math teachers moderated by Cindy Bryant, veteran teacher, former NCTM Board Director, Presidential Awardee, and former state K–12 Math Director in Missouri. Share ideas, best practices and what works for you with your peers to help every student unleash their potential.  Contact Cindy.

September Webinar!

Topic: Cultivating Intellgience and Talent through a Growth Mindset

Time: Wed, 9/24 at 2 p.m. ET

Mindset – a simple idea discovered by world-renowned Stanford University psychologist Carol Dweck can determine whether a student focuses on improving rather than how smart they are. This session will focus on promoting a growth mindset that fosters student motivation, productivity, and the learning of mathematics.

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