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Strands of Mathematical Proficiency

Think about your gut-response to this statement:  Anybody can do math.  There was a point in my life when I would have vehemently objected to the truth of that short sentence.  And I would have readily said that anybody most certainly did not include me.  Then, very shortly after I began teaching, I had to teach math – or find another job.  Luckily, my principal paired me with a mentor who was an incredibly gifted math teacher herself.  I not only learned how to teach math, I learned how to make sense of  math.  I developed, in today’s verbiage, a mathematical mindset.

A mathematical mindset is a way of thinking about and doing math that is flexible, active and has at its core the idea that math makes sense.  This mindset is grounded in knowing that math is much more than just a collection of rules and procedures that must be memorized; it is developed through wrestling with mathematical ideas, solving problems, and making connections.  In my experience, helping learners develop a mathematical mindset is crucial to them becoming proficient in mathematics.

According to Adding It Up, the five strands of mathematical proficiency are:  (1) Strategic Competence, (2) Conceptual Understanding, (3) Procedural Fluency, (4) Adaptive Reasoning, and (5) Productive Disposition. As teachers, we talk a lot about the first four.  They’re even embedded in our content and process standards.  But I wonder what might happen if we intentionally paid more attention to and planned for helping learners grow a productive disposition towards math.  I suspect they’d develop mathematical superpowers!  We could empower them to tackle math problems head-on and struggle through tough work, coming out of it feeling like super heroes.

Which brings us to the question of how to accomplish this. The very first place to start is with ourselves.  Ask yourself these questions:  Do I approach mathematics with a growth mindset?  Do I think anyone can learn mathematics, including myself, and recognize that effort and struggle is an important part of learning mathematics? 

Once we establish our starting point—that we have a growth mindset about mathematics ourselves—we can begin to think how we might structure our classrooms to promote mathematical mindsets.           
A key step in helping students develop a productive disposition towards mathematics is to select mathematical tasks that are worthwhile.  Worthwhile tasks have clear mathematical goals, provide opportunities for discussion, provoke thinking and reasoning about mathematics, and engage students in doing mathematics.

Let’s compare two tasks.   

Task 1:
Making Change
You have 1 quarter, 2 dimes, and 2 pennies.  How much money do you have?

Task 2:
Making Change
Make 47¢ using exactly 6 coins with either quarters, dimes, nickels, or pennies.

Make 47¢ in three different ways with either quarters, dimes, nickels, or pennies.  How can you show your answer using pictures, numbers, and words? (Thad Domina and Robert Kaplinsky)
Task 1 is a traditional mathematical task we might see in a classroom when students are learning to count money.  Task 2 addresses the same content but in a very different way.  How does Task 2 promote thinking and reasoning about mathematics in a way that Task 1 does not?  Which task would result in the most student dialogue about mathematics?

When planning to include worthwhile tasks in math instruction, you may begin by tweaking problems in your textbook or other instructional materials. 

There are also many places to look for tasks.  Here are some FREE resources: These three books by authors John A. Van de Walle, Jennifer M. Bay-Williams, LouAnn H. Lovin, and Karen S. Karp aren’t free, but they’re great resources.  They’re arranged by grade band, and I have found them a valuable resource.
  • Teaching Student-Centered Mathematics:  Developmentally Appropriate Instruction for Grades PreK-2 (Volume I, 3rd Edition)
  • Teaching Student-Centered Mathematics:  Developmentally Appropriate Instruction for Grades 3-5 (Volume II, 3rd Edition)
  • Teaching Student-Centered Mathematics:  Developmentally Appropriate Instruction for Grades 6-8 (Volume III, 3rd Edition)
As you plan for implementing more problem-based tasks in your math instruction, consider the checklist in the table below.  It will help you make decisions about which tasks you want to use.

Source:  From Chapter 2:  Teaching Mathematics through Problem Solving in Teaching Student-Centered Mathematics:  Developmentally Appropriate Instruction for Grades PreK – 2  (Volume I, 3rd Edition) Authors:  John A. Van de Walle, Jennifer M. Bay-Williams, LouAnn H. Lovin, and Karen S. Karp, © 2016









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