Problem Solving: An Approach to Understanding and Critiquing Our World
July 2022
Problem-solving and critical-thinking skills prepare students to understand and critique their world, preparing them to be contributing leaders in society. They are skills that all individuals need in decision-making, in the workforce, in their communities, and in daily life. “All children have remarkable abilities to learn substantial mathematics when provided mathematics learning opportunities that emphasize sense making and problem solving” (NCTM 2020a, p. 28).
Many aspects of problem solving are valuable. In PK–12 classrooms these include but are not limited to tasks and problems that have multiple entry points and varied solution strategies and that consider the role of context and multiple representations along with cultural and community connections. A classroom environment that promotes problem solving provides opportunities for creativity, supports developing a positive mathematics identity for all learners, and provides a pathway for developing a deep understanding of mathematics.
Hypatia of Alexandria, Egypt, known as one of world’s first female mathematicians, often traveled with her father in his work. When she had a wondering, he would regularly respond that the answer was math, and thus began her investigations. As a result, she fell in love with mathematics. Goris (2017) wrote about Hypatia in a series of STEM-focused books. My favorite quote is, “Hypatia learned to ask big questions and ponder big answers.” We want all our students to ask those big questions and ponder those big answers as they engage in problem solving.
NCTM has advocated for problem solving for decades. From 1980 in the Agenda for Action, which included as its first recommendation that mathematics curriculum have a focus on problem solving; to the release of NCTM’s Curriculum and Evaluation Standards (1989), reminding us that problem solving is a process in which developing students as problem solvers is essential if students are to become productive citizens, develop confidence in using mathematics meaningfully, and be able to model real-world problem situations. Problem solving was further emphasized in NCTM’s Principles and Standards for School Mathematics (2000), focusing on building new mathematical knowledge, solving problems that arise in mathematics and other contexts, and applying and adapting appropriate strategies to solve problems. Later in NCTM’s Principles to Actions: Ensuring Mathematical Success for All (2014) and the Catalyzing Change series, (2018, 2020a, 2020b), a focus on problems solving was seen as an essential part of effective equitable instruction.
And yet, even with this focus for decades on problem solving and acknowledging its importance and value, we still have been challenged to fully embrace problem solving at times, often finding a continued focus on learning procedures, a lowering of expectations for some students, and limited access to instructional resources to support problem solving. Furthermore, because of the impact of high-stakes assessment in states/provinces and districts, skills and facts are often emphasized rather than problem solving, critical thinking, and reasoning (NCTM 2014). We must address these challenges and barriers to ensure equitable access to mathematical problem solving for each and every student.
Why should we include a problem-solving approach to learning and exploring mathematics? A few reasons include that problem solving—
- supports making connections across disciplines;
- prepares students for future professional opportunities;
- develops students’ positive mathematical identity;
- is a matter of equity and access;
- builds students’ confidence, persistence, flexibility, creativity, perseverance, and curiosity;
- gives students voice and promotes discussion;
- shifts math authority to students; and
- has a positive effect on learning.
(NCTM 2000; 2014, 2018, 2020a, 2020b)
Both mathematical thinking and creativity are key parts of problem solving because “children experience joy when they are encouraged to be creative and are provided with mathematical choices, such as generating their own approaches and strategies for solving mathematical problems” (NCTM 2020a, p. 21). A problem-solving approach to instruction provides a learning environment that promotes thinking mathematically and provides a space for nurturing creativity and innovation. It is essential that we offer students the resources and space for being creative through multiple opportunities for students to examine problems and issues of interest to them, that have meaning in their lives, and that help them makes sense of their world. (NCTM 2018, 2020a, 2020b). We as teachers can and must intentionally facilitate this kind of environment, providing needed resources and supports to each and every student to nurture creativity and mathematical thinking.
Recently Guadalupe Siclia, a grade 3 teacher at Gardens Elementary School in Pasadena, Texas, shared with me, “It's important to me that my students engage in problem solving and problem-solving-related tasks because I want to help build a generation of mathematicians that are able to think, discuss, and persevere when working through problems that can be challenging.” Powerfully stated! A classroom environment and culture of problem solving supports all students in their learning as thinkers and doers of mathematics.
Approaching learning mathematics that includes a focus on problem solving is exciting, rewarding, and essential. What is your plan for preparing students to understand and critique their world through problem solving? Let’s look for opportunities for supporting all PK–12 students on their journey in problem solving!
Trena Wilkerson
NCTM President
@TrenaWilkerson
References
Goris, Brittany. 2017. Hypatia: Explorer of Geometry. Seattle, WA: Girls Rock Math.
National Council of Teachers of Mathematics (NCTM). 1980. An Agenda for Action: Recommendations for School Mathematics in the 1980s. Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 1989. Curriculum and Evaluation Standards for School Mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 2018. Catalyzing Change in High School Mathematics: Initiating Critical Conversations. Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 2020a. Catalyzing Change in Early Childhood and Elementary Mathematics: Initiating Critical Conversations. Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 2020b. Catalyzing Change in Middle School Mathematics: Initiating Critical Conversations. Reston, VA: NCTM.