The Art of Teaching and Learning Math

By Brittany Puffenberger, Math Teacher

When I was a student, I was taught that you are either a “math person” or you’re not. These “math people” were known as left-brained thinkers and were often the boys in the class. The right-brained thinkers, often the girls in the class, were known to be better in creative subjects, like the arts. We were pigeonholed into these roles. We saw these as our identities.

Former math pedagogical practices cater to the “left-brained thinkers.” Under these practices, it’s commonly believed that math is a series of formulas to remember and regurgitate to arrive at a correct solution. But math is so much more than this! It wasn’t until I became an elementary teacher that I came to this realization. It requires skill, creativity, and flexible thinking.

Learning mathematics is an art. Teaching mathematics is an art.

An effective problem-based lesson structure requires not only intentional and strategic moves, but also productive struggle and revision of thoughts as a student. This type of structure allows for discourse, equitable access for all, and a means for multiple representations.

The art of learning mathematics can be compared to the five stages of the artistic writing process. The eight Standards for Mathematical Practice are visible and referenced in this process:

1. The Prewrite: When composing a piece of writing, students must decide what graphic organizer to use to sort out their thinking. For example, if they’re asked to write a comparison piece, a strategic choice to organize thoughts would be a Venn Diagram. Compare this to a math task given. Students first need to make sense of the abstract problem (CCSS MP1 & CCSS MP2) and determine which tools or representations they will use to model and solve it (CCSS MP5). Will they show their thinking using a number line, a diagram, or physical manipulatives? The creative expression opportunities are boundless.
2. The Draft: Students share their thinking with a peer or small group. Tier 1 and Tier 2 vocabulary words may be used informally to describe their thinking before formally introducing them to a Tier 3 content-specific vocabulary word later in the process. Mistakes are made and even encouraged. These are learning opportunities.
3. The Edit: Compare this to a peer edit. Here, learners can listen to other classmates who have chosen to approach problems differently. They evaluate their thoughts in comparison with their own. They critique work and take feedback from others and may even see the problem in a way they may have never imagined before (CCSS MP3).
4. The Revision: The mathematical practice standards CCSS MP7 and CCSS MP8 are present in this step of the artistic process. After students have seen multiple means of expression, they may then revise their thinking. An example: when subtracting a fraction from a mixed number such as 1 3/8–7/8, changing the mixed number to an improper fraction is the common solution. But is this the only way? Could they use a number line to count from 7/8 up to find the difference? To many, this is a novel strategy. How could they get there? Perhaps when observing another classmate try this strategy, they realize that their thinking can be flexible, and shift based on the problem given. They may also make the connection in seeing the relationship between addition and subtraction.
5. The Final Publication: The culmination is students explaining their thinking both verbally, and in writing, while attending to precision (CCSS MP6). They are now using Tier 3 vocabulary and have conceptualized the learning goal. This allows them to go out into the world and apply their learning in real scenarios (CCSS MP 4). They’re better equipped at authentic problem-solving and continue to work towards developing procedural fluency.

The art of teaching mathematics requires teachers to use the Five Practices for Orchestrating Productive Mathematics Discussions. The art lies in guiding students in their discovery rather than acting as the holder of all knowledge.

Like a seasoned musician that can play a tune by ear, effective teachers can anticipate the strategies students may use in solving a problem as well as the common misconceptions they may have. They actively monitor and listen for these procedures and common errors then deliberately select students’ work to display (even those who have not arrived at a correct solution). The sequence in which responses are shared should be planned and thought out for maximum learning impact. Let’s say the end goal is for learners to be able to write expressions to find the volume of a rectangular prism. A teacher might first choose to showcase a learner who used prior knowledge by counting the number of cubes in a layer and multiplying it by the number of layers. This could then be connected to another student’s idea who decomposed the area of the base into its factors’ side lengths. What the teacher is doing is illustrating the connection between finding the volume of a rectangular prism using base x height or length x width x height. Two approaches; one result.

Mastering any form of art doesn’t happen all at once. It takes time, practice, continuous learning, and self-reflection to cultivate this craft. If implemented with integrity, a problem-based math classroom will encourage all our students to see themselves as “math people.”

Brittany Puffenberger has over ten years of experience in grades K-6 in both brick-and-mortar and cyber classrooms. She currently teaches 5th grade at a cyber charter school in Pennsylvania. Brittany has a Bachelor of Science Degree in Elementary Education from Slippery Rock University and a Master of Education Degree in Education Policy, Organization, & Leadership from the University of Illinois at Urbana-Champaign. Her greatest interest in the field is studying more effective, inclusive, and innovative instructional practices to provide equitable outcomes for all learners.

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