When talking about the teaching and learning of mathematics, we often bring our own experiences to the conversation. For example, if you experienced mathematics as an endless list of problems that you had to solve in order to arrive at a single correct answer, that is likely how you think about mathematics in your own classroom.

There is a place for problem solving and calculating correct answers. However, it is misguided to reduce mathematics teaching and learning to a series of steps to be performed to yield a numerical answer.

That outdated approach takes away from the potential thinking, creating, discussing, debating, and exploring of mathematics.

When we developed *The Mathematics Playbook*, our focus was on translating what works best in the mathematics classroom to foster, nurture, and sustain Visible Learners. To do this, we need to first check our own understanding of what it means to learn mathematics, so that we can create classrooms that promote the type of thinking and learning that deepens understanding of mathematics concepts and skills. At the same time, these concepts and skills must be usable, durable, and flexible in solving authentic and engaging mathematics tasks.

While addressing all of this in a single blog is not possible, we do want to highlight the essential question for teaching and learning in mathematics. That’s exactly the focus of this Corwin Connect blog. What is mathematics learning, really?

**The Elements of Mathematics Learning**

Mathematics learning has five elements that should be a part of every teaching and learning experience. Those elements are:

- conceptual knowledge
- procedural fluency
- strategic competence
- adaptive reasoning
- productive dispositions

Let’s look at each of these separately to see where we can enhance the learning opportunities in our own classrooms.

*Conceptual Knowledge. *A concept is an abstract idea that serves as the fundamental building block for principles, thoughts, and beliefs (Goguen, 2005). In mathematics, concepts are mental representations of mathematical ideas such as equivalence, non-standard units, percentage, fractions, ratio, division, and cardinality. E. Clark (1997) defined a concept as a big idea that helps make sense of, or connects, lots of ideas. Thus, conceptual knowledge is the relational understanding of these concepts in such a way that learners know the significance of the concept and how this concept is a part of specific content, skills, and understandings.

Take a few moments and reflect on your own standards, curriculum, or learning experiences. What mental representations of mathematical ideas must your students know and understand?

*Procedural Fluency. *If conceptual knowledge is all about big ideas, then procedural fluency is the knowledge of what procedures to use, when to use them, and how to use them (National Research Council, 2001). Notice that procedural fluency is more than just memorizing and executing procedures, algorithms, and steps. Again, procedural fluency is relational understanding of procedures, algorithms, and steps. The “what” of procedural fluency is dependent on conceptual knowledge. Conceptual knowledge helps us understand the when and how of procedures, algorithms, and steps.

When procedural fluency is an essential part of our mathematics teaching and learning, our students understand the rules and the reasons. Whether find the sum of two numbers, the area of a triangle, determining the set of solutions to an inequality, or determining the related rate, learners demonstrate their procedural fluency when they can utilize specific procedures, algorithms, and steps with flexibility, accuracy, and efficiency.

Learners must use procedures, algorithms, and steps flexibly in a mathematics problem. For example, if the two fractions have unlike denominators, learners must find the least common multiple (LCM) so that they can find a common denominator. Learners must also be accurate in their approach to solving a mathematics problem. Using a specific algorithm to find the sum of two fractions with unlike denominators still requires that they accurately find the LCM and be accurate in the computation. Finally, learners must know how to use a procedure efficiently so that they do not consume too much working memory. For example, if the sides of a triangle represent a Pythagorean triplet, learners should recognize this and use their conceptual knowledge to more efficiently determine congruency.

*Strategic Competence.* Conceptual knowledge and procedural fluency, alone, do not reflect mathematics learning. For example, consider the concepts of a unit circle and trigonometric ratios. At their core, these concepts are associated with procedural fluency in calculating the angles of a triangle or solving trigonometric equations. However, mathematics proficiency must include the capacity of the learner to recognize when these two concepts and their associated procedures are applicable to a given context. Consider the two problems below and see if you can recognize which one requires strategic competence:

**Problem #1:**Find the area of a rectangle of with a length of 28 cm and a width of 4 cm. What is the perimeter of this rectangle?**Problem #2:**The school football field is 65 yards by 35 yards. If the players run around the field four times in physical education, what total distance did they run during the class period?

Problem #2 requires learners to first represent the problem using numbers, symbols, or models/visual representations. From there, learners must identify the relationships within the problem to develop an approach for solving the problem. The capacity for learners to identify these relationships moves them away from simply grabbing numbers and plugging them into a formula. Instead, learners are more flexible and efficient in utilizing specific procedures to arrive at a mathematically accurate solution.

Strategic competence pulls together conceptual knowledge and procedural fluency. Strategic competence relies on relational understanding of the mathematics. Proficiency here is demonstrated by learners’ ability to formulate, represent, and solve mathematics problems (National Research Council, 2001).

*Adaptive Reasoning. *This element of mathematics learning—a part of what it means to be proficient in mathematics—zeros in on the thinking of learners. Learners must not only have the conceptual knowledge and procedural fluency to demonstrate strategic competence, but learners must think logically to perform the following (Battista, 2017; Darwani et al., 2020; Muin et al., 2018):

- unpack the details of the problem,
- discern the relevant from the irrelevant details,
- reason through the approach to solving the problem,
- consider alternatives, and
- justify and explain the approach and solution.

Take a few moments and look back over the examples presented in this blog. Where do you see adaptive reasoning playing out in those problems? How about in your own classroom?

*Productive Dispositions*. The final element of mathematics learning brings us face to face with learners’ general disposition toward mathematics. Do learners find mathematics relevant? Do learners see value in mathematics? Finally, what are learners’ attitudes toward mathematics and toward themselves as mathematics learners? Yes, we are talking about attitudes, beliefs, engagement, and efficacy. Productive dispositions develop alongside the other elements of mathematics learning, and we can be intentional, purposeful, and deliberate in developing these dispositions. For example, we can focus on the disposition of persistence in problem solving during a particular unit. Maybe we focus on the usefulness of mathematics during a unit that integrates science. The access and opportunity to engage in the highest level of learning possible enhances the productive dispositions of learners in mathematics. Our conversation regarding productive dispositions is a perfect segue into our blog. Learners’ attitudes, beliefs, engagement, and efficacy are integral parts of their mathematics learning journey. Furthermore, this is where we play a major role in that journey.

Teaching and learning mathematics involve much more than just calculation. To allow our students to experience the richness of mathematics learning, we must ensure that our mathematics teaching is rich with all of the elements that make up mathematics learning. So, let’s get to it!

**References**

Battista, M. (2017). Mathematical reasoning and sense making: Reasoning and sense

making in the mathematics classroom, grades 3-5. National Council of Teachers of

Mathematics, pp. 1–22.

Clark, E. (1997). Designing and implementing an integrated curriculum: A student-centered

approach. Holistic Education Press.

Darwani., Zubainur, C. M., & Saminan. (2020). Adaptive reasoning and strategic

competence through problem-based learning model in middle school. Journal of

Physics: Conference Series, 1460(1), 012019.

Goguen, J. (2005). What is a concept? Conceptual structures: Common semantics for

sharing knowledge. Lecture Notes in Computer Science, 3596, 52–77.

Muin, A., Hanifah, S. H., & Diwidian, F. (2018). The effect of creative problem solving on

students’ mathematical adaptive reasoning. Journal of Physics: Conference Series,

948, 012001.

National Research Council. (2001). Adding it up: Helping children learn mathematics. In J.

Kilpatrick., J. Swafford, & B. Findell (Eds.), Mathematics learning study committee,

center for education, division of behavioral and social sciences and education.

National Academy Press.