When the results of U.S. national mathematics assessment (NAEP) were released earlier this school year, they confirmed what most educators already knew — that students would not do as well as they normally would on their mathematics assessment. If we had not seen this decline in test scores, I think it would have raised serious concerns about the validity of the NAEP test itself. With students, teachers, school administrators, parents and other stakeholders facing numerous challenges in the years leading up to the 2022 NAEP test, I am certain that most of these stakeholders would agree that students, in general, were left with more academic needs than any time in recent memory. The open question remains – what can we do to ensure that students learn the mathematics that they need to be college and career ready? The immediate answer is that we need to focus on what is best for all students. To do so means striking a balance between conceptual understanding and procedural fluency when teaching mathematics to students.
The National Council of Teacher of Mathematics (NCTM) stated that 2022 NAEP results “point toward the need for a greater focus and higher priority on mathematics instruction.” Shifting the focus away from test scores and towards a focus on mathematics instruction is a first step toward enacting positive change for students. Current NCTM President, Kevin Dykema, reminds us that “students are capable of learning and making sense of mathematics.” A positive learning experience coupled with research-informed and equity-based instructional strategies increase learning and students’ belief in their ability to do mathematics.” We know that successfully applying these strategies in the classroom will yield powerful results for student learning. But we also know that many teachers will need support to do so successfully. When the 2019 NAEP was released, NCTM President, Robert Berry III, made an observation that still rings true today. He said that “we continue to see learning time dominated with test preparation rather than rigorous and engaging instruction.” As a high school mathematics teacher, I have seen firsthand that we need to focus on what students need. I believe that this focus needs to be rooted in the purposes of high school mathematics.
NCTM’s Catalyzing Change speaks directly to these purposes of mathematics, stating that “high school mathematics empowers students to — expand professional opportunity; understand and critique the world; and experience wonder, joy, and beauty.” Another mathematics teacher, Kanchan Kant, has blogged that she has resolved herself not to get “bogged down by the need for tests and quizzes” and, instead, “teach via exploration, to focus on inquiry more than the procedure.” In my teaching and in my own learning of mathematics, exploration is at the heart of the wonder, joy, and beauty of mathematics. By teaching using exploration and inquiry in mathematics, we help to ensure that students are doing mathematics for their intended purposes. Prior knowledge and skills that students need to explore can be woven into exploration and inquiry lessons. Programs like CollegeReadyMath can be used to create customized learning paths for each individual student to help ensure successful exploration and inquiry as well as helping students to achieve high school, college and career readiness. In this sample lesson, CollegeReadyMath makes connections between linear equations and slope by using graphs. Using tools like this can give students an opportunity to explore concepts before, during, or after instruction to help bolster conceptual understanding and procedural fluency.
We know that “teachers who make a practice of building fluency from conceptual understanding routinely connect conceptual understanding with procedural fluency so that students can make meaning of the mathematics and develop a positive disposition toward mathematics” (Catalyzing Change, NCTM). Developing this positive disposition towards mathematics can help to achieve equitable results in the classroom. If developing conceptual understanding alongside procedural fluency helps accomplish this, then it is definitely a great place for teachers to focus on their instruction. In her blog posting, Jennifer Knudsen, reminds us that resources like Mathematical Argumentation in Middle School (Corwin) are available to teachers as a tool to help teach their students to think like mathematicians and to use both their understanding of concepts and knowledge of procedures to solve problems. A few years ago, in a talk at an NCTM Annual Meeting, Juli Dixon, described the “five ways we undermine efforts to increase student’s achievement.” She also provided helpful tips and advice for how to avoid these common pitfalls. I mention this here because “neglecting opportunities to connect concepts and procedures” was included as one of the five ways. Since that talk, she has made examples available of how to be intentional and explicit about making the connections between concepts and procedures. By helping teachers focus on improving mathematics instruction, including making connections between concepts and procedures, and by developing positive student dispositions towards mathematics, we can help all students to be more successful and productive learners of math. This, in turn, will help ensure that students perform better on future assessments of mathematics.