Mathematics Proficiency Strands

[Nathen Paisley]

AddingIt Up explores how students in pre-K through 8th grade learn mathematics and recommends how teaching, curricula, and teacher education should change to improve mathematics learning during these critical years.

The committee identifies five interdependent components of mathematical proficiency and describes how students develop this proficiency. With examples and illustrations, the book presents a portrait of mathematics learning:

The committee discusses what is known from research about teaching for mathematics proficiency, focusing on the interactions between teachers and students around educational materials and how teachers develop proficiency in teaching mathematics.

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The proficiencies of Understanding, Fluency, Problem Solving and Reasoning are fundamental to learning mathematics and working mathematically and are applied across all three strands Number and Algebra, Measurement and Geometry, and Statistics and Probability.

Fluency describes students developing skills in choosing appropriate procedures, carrying out procedures flexibly, accurately, efficiently and appropriately, and recalling factual knowledge and concepts readily. Students are fluent when they:

Problem-solving is the ability of students to make choices, interpret, formulate, model and investigate problem situations, select and use technological functions and communicate solutions effectively. Students pose and solve problems when they:

Reasoning refers to students developing an increasingly sophisticated capacity for logical, statistical and probabilistic thinking and actions, such as conjecturing, hypothesising, analysing, proving, evaluating, explaining, inferring, justifying, refuting, abstracting and generalising. Students are reasoning mathematically when they:

Information Communication Technologies (ICT) are powerful tools that can support student learning. Students can develop and demonstrate their understanding of concepts and content in Mathematics using a range of ICT tools. It is also important that students know how to use these ICT efficiently and responsibly, as well as learning how to protect themselves and secure their data.

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How do you define mathematical proficiency? Had I been asked this question early in my teaching career, I might have defined it as the ability to solve problems efficiently and accurately. In the back of my mind, I might have thought it meant having good number sense for procedural computations or equation solving. The five strands of Mathematical Proficiency are conceptual knowledge, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.

Honestly, as a student, I was really good at memorizing and following procedures. I excelled in math the way I was taught. Through lectures and endless procedural practice. Therefore, my personal math experiences informed my definition of mathematical proficiency and my initial instructional style. Luckily, I do love to learn! As a novice teacher, I read several articles and books and attended professional learning opportunities over the years. These outlets enhanced my view, definition, and understanding of mathematical proficiency.

To be proficient in mathematics means that one learned mathematics successfully. But what is the threshold for that success? What skills does a student possess and demonstrate that deems them mathematically proficient? I pursued professional learning opportunities seeking answers to these questions.

One impactful professional learning workshop I attended shared some research from the National Research Council. At the end of 1998, the National Research Council established the Committee on Mathematics Learning including a diverse group of math education experts from across the country.

The National Council of Teachers of Mathematics (NCTM) references the Committee on Mathematics Learning definition of procedural fluency, as does our own K-5 Texas Essential Knowledge and Skills (TEKS) Math Standards.

The grant resulted in the creation of Research-Based Instructional Strategies (RBIS). Now Texas math classrooms incorporate RBIS, a set of practices supported by research, regardless of instructional materials. The RBIS are based on the science of how students best learn math in K-12 classrooms.

RBIS 1 is the balance of conceptual with procedural. This means we should pursue rigor by balancing conceptual understanding, procedural skill, and fluency, and apply this balanced understanding to mathematical applications as required by the standards in the TEKS.

Conceptual knowledge and procedural fluency are only two of the strands of mathematical proficiency. To promote mathematical proficiency in students, these two strands are only the beginning of what we should develop in our students and seek out in our instructional resources.

(B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

(C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

I would argue that if we are truly planning our lessons to provide intentional opportunities for students to develop and demonstrate the TEKS Mathematical Process Standards, then we are focusing on developing mathematically proficient students.

I began to research what I could about developing a productive disposition, and that is when I was introduced to the concept of growth vs. fixed mindset. I read early articles from Dr. Carol Dweck, who defined mindset and its impact on learning. She later detailed this research in her book Mindset: The New Psychology of Success.

My students who stated they were never good at math held a fixed mindset. Their previous negative experiences in math made them view their failure in math as part of their identity. Therefore, these students did not believe that they could ever be successful in mathematics. This belief impacted their ability to try, to care, or to learn. I knew I needed to foster a growth mindset environment in my classroom that supported the belief that effort creates ability.

Hiebert, James, Thomas P. Carpenter, Elizabeth Fennema, Karen C. Fuson, Diana Wearne, Hanlie Murray, Alwyn Olivier, and Piet Human. 1997. Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann.

It is the stated goal of every teacher across the country to support all students in achieving mathematical proficiency. This is key, in helping to develop students who feel confident and proficient in using their math skills to solve real-life challenges once they move past their classroom and school years.

Procedural fluency is the knowledge and use of rules and procedures used in carrying out mathematical processes and also the symbolism used to represent mathematics. For example, skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

Adaptive Reasoning is the capacity of a student to demonstrate logical thought, reflection, explanation, and justification when it comes to problem-solving. This capacity to reflect on their work, evaluate it, and then adapt, as needed, is the adaptive reasoning.

The development of student mathematical proficiency takes time. In each grade, students need to make progress along every strand mentioned above. Each strand is important and interwoven with the others. Additionally, the last three of the five strands develop only when students have experiences with solving problems as part of their daily learning in mathematics.

Because the issue of student proficiency and confidence in math is such a central factor in providing a successful math instructional program, TKL has put together many resources for teachers and schools to leverage.

Here in 2021, we are all waiting and hoping for the higher ups to realize that this year is not a year to give our students the standard state assessments. Whether or not that happens, as educators we do still wonder if our students are proficient.

I personally love the description of mathematical proficiency put forth in the book Adding it Up: Helping Children Learn Mathematics. On page 5 of the book they show a picture of the 5 strands of mathematical proficiency.

And so as you are starting to worry about if your students are mathematically proficient this year, I want to encourage you to keep these 5 things in mind. Yes they are a lot harder to quantify than the score a kid gets on a state assessment, but they are so much more meaningful.

The Western Australian Curriculum features v8.1 of the Australian Curriculum for mathematics. Western Australian schools may continue to use the previous version (available in the PDF below) for teaching, assessment and reporting purposes for 2016.

The proficiency strands Understanding, Fluency, Problem Solving and Reasoning are an integral part of the mathematics content across the three content strands: Number and Algebra, Measurement and Geometry, Statistics and Probability. The proficiencies reinforce the significance of working mathematically within the content and describe how the content is explored or developed. They provide the language to build in the developmental aspects of the learning of mathematics.

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