Missing pieces in math education

Analysis of the history of U.S. mathematics education demonstrates two key facts: The mathematical knowledge and pedagogy of teachers is crucial in early grades, and the scope and sequence of mathematics in high school is not adequate but is steeped in tradition.

The history of mathematics education in the 20th century shows a continual “pendulum swing” between drill and practice and discovery learning in all grades (Walmsley, 2007). Continually, students were either exposed to procedures and focused on memorization of facts, or they were asked to engage with mathematics in their own way, often without understanding basic math. Of course there are exceptions to this, but generally, we tend to pivot back and forth in these pedagogies. For best results, students need both instruction in mathematical basic facts and procedures and opportunities to explore mathematical ideas.

The National Council of Teachers of Mathematics made great strides in 2000 with the release of the Principles and Standards for School Mathematics. This comprehensive set of standards describes the content that students at all levels should learn. It also promotes pedagogy where students can learn mathematics with true understanding but also develop the ability to compute efficiently. Despite these efforts, we still see a lack of some content and a dislike of mathematics in our schools and students.

The role of teachers

For many students, their experiences with mathematics (and ultimately their opinion of it) depend on their teachers’ familiarity with mathematics. As numerous articles in the December/January issue of Kappan note (Jakopovic, 2024; McClam & Cruz, 2024), many teachers responsible for getting students off to a strong start are uncomfortable with math. Having worked with many preservice teachers in my career, mathematics is often the field that elementary teachers are the most uncomfortable with. And as is so often the case in education, teachers who are unfamiliar or uncomfortable with a topic will revert to the way they were taught.

While I have seen amazing elementary teachers in mathematics classrooms, I’ve also listened to elementary teachers tell me they don’t feel confident teaching anything besides the basics in mathematics. Some have told me they decided to major in elementary education because it had the least rigorous or lowest-level mathematics requirements for a degree.

Today, many elementary programs continue to require little mathematical knowledge as a foundation of their program (maybe college algebra or a “math for elementary teachers” course). Therefore, unless we push for more knowledge and confidence in mathematics in our elementary training, we may see a continuous cycle of rote learning or a focus only on the basics of arithmetic in the early grades. Equally important is the need for teachers to positively promote mathematics in early grades so that all students feel confident and happy engaging in mathematics.

The problem with the current sequence

The real issue with the mathematics curriculum in the early grades is that we need to provide students with the opportunity to explore. From the beginning, we teach children that there is one right answer in mathematics. And, unfortunately, some teachers also teach only one method to that answer.

We need to encourage students to ask questions, explore the data surrounding that question, and then decide on the best answer. This is where data and statistics come in. In the “real” world, the average adult is going to use data analysis and interpretation much more than a classic advanced topic like geometry or calculus. This engagement with data should begin in elementary school.

The December Kappan explores the importance of student engagement and real-world thinking in math (see, for example, Parker-Holliman & Marcel-Herbert, 2024), but it doesn’t address the limitations of the current math sequence. We tend to teach primarily arithmetic and number theory in the early years; this is then followed by our classic track of one year of Algebra I, one year of Geometry, one year of Algebra II, one year of Precalculus, and then one year of Calculus for those students who were advanced enough in 8th grade to take Algebra I.

This creates two major problems. The first is that a student who does not take 8th^_grade algebra never gets the opportunity to take calculus in high school. Even if they are capable or end up really enjoying mathematics, a decision they made in 7th grade ends up having long-term consequences. Students who apply to colleges in STEM fields may miss out because they didn’t reach the calculus level in high school.

The second problem is that we are not naturally integrating mathematics principles into our curriculum because we compartmentalize our subjects at the high school level. We are one of the only countries in the world to do so. With our current sequence, the important subject of statistics often gets completely neglected or takes “second place” when a student is trying to fit in the traditional sequence in their schedule.

Proposing a different approach

A much better model would be to end the compartmentalization of math topics. Each year of high school math should include statistics, probability, and data science; algebraic principles; number theory; and geometric and trigonometric principles. While some textbook publishers and schools have tried this approach, it is not popular. The historical mathematics track is deeply embedded in high schools — and colleges continue to expect to see that sequence of courses on transcripts. I heard a presentation by a major textbook author one time who said the publisher told the authors they could put whatever they wanted in the actual books, but the titles had to be Algebra I, Geometry, and Algebra II so that they would sell.

Where do we go from here? We must push faculty leading elementary teacher training to include rigorous mathematics courses in their programs. We also need to support elementary school teachers in the teaching of mathematics so that they not only become confident in the subject but also express the creativity and love of mathematics to their students.

Regarding content, it will be extremely difficult to break the track as established in most high schools. But teachers can integrate other fields of mathematics into those traditional courses as much as possible. Specifically, we need to start to integrate statistics into mathematics at all levels. We live in a data-driven technological society, and the integration of statistics is paramount for future citizens.

References

Jakopovic, P.M. (2024). Math narratives and autobiographies. Phi Delta Kappan, 106 (4), 20-24.

McClam, R. S. & Cruz, R.A. (2024). Beyond the math wars: Focus on teachers to improve instruction. Phi Delta Kappan, 106 (4), 8-13.

Parker-Holliman, N. & Marcel-Herbert, T. (2024). Everyone is a math person. Phi Delta Kappan, 106 (4), 26-30.

Walmsley, A. (2007). A history of mathematics education during the 20th century. University Press of America.