As a university professor in natural sciences, I have come to a generalization about how I think K-12 math education too often fails in reaching its primary goals. My focus is broad, extending from basic arithmetic through calculus. Not all teaching and learning follows this generalization, and although I think the system is not completely broken, I am convinced that understanding the context of my generalization would illuminate errors in our approaches and would suggest directions we can move so that more students become proficient in math.

Consider how we usually teach and learn multiplication. Right
at first, students might be introduced to the concept of adding up the members
of some groups of equal size (like, 4+4+4 is the same as 4 times 3), using
counting as the fundamental proof. After that, training is heavily dominated by
memorization of tables and repeatedly writing the answer to a mix of simple exercises
in recall. Now, don’t get me wrong: Repetition can be great at reinforcing a
skill, making it a necessary component of good math education. Yet, by
weighting too strongly on repetition and drills, the concept gets buried under
a mass of scripted rubble, as the system collapses in fulfilling the needs of a
student who loses grasp of the central point.

In more advanced mathematics, such as trigonometry, K-12 education
typically follows the same general pathway. Teachers and textbooks give token time to the
fundamentals that lead to some math fact, including providing proof. Then, they
highlight the conclusion that comes from that proof. Students memorize that
conclusion as a rule, and then as with multiplication, they drill-drill-drill. Homework
is nearly always dominated by carrying out the tasks of repetition, and exams,
for the most part, assess whether students can carry out the same assigned
tasks. With insufficient emphasis on the fundamental reasoning in the practice
and assessment chain, students themselves naturally focus on learning the rules
and carrying out the tasks, glossing over the motivations behind it all. The
supporting proof and fundamental understanding gets buried in the quest to
complete long lists of exercises, while the student and the instructor often see
illusions of understanding based on success in the exercises, or students fall
behind with no clear path to understanding. Over the whole trigonometry course,
students memorize chains of rules that they then usually forget within a few
weeks as they move to other topics. Yet, deep understanding of trigonometry,
like most branches of mathematics, requires remembering only a handful of basic
fundamentals, from which nearly all rules can be easily derived, often by
simply drawing a triangle. Students who learn by following the public school paradigm eventually
find themselves lost in the sea of rules, frequently looking up the rules again
and again in order to carry out future tasks. The core of the students’ understanding of the math itself is therefore shallow. Many students who are perceived as successful
complete assigned tasks and get excellent grades on exams, but really
understand little of the subject.

An appalling fraction of former high school students who have
completed even our best courses in Advanced Placement calculus, who then take calculus again in
college, fail to attain a C or higher there. Calculus instructors in college
frequently question why students who seem so well prepared on paper are
actually poorly prepared for the intuitive thinking of a good college calculus
course. In contrast, students who succeed typically have gained fundamental understandings
of the roots of the algorithms applied: They are not simply satisfied by
completing the drill or memorizing the rule. Understanding the motivation for
the rule is far more important than the rule itself. Flaws driven by shallow
understandings accumulate over time, eventually generating an overwhelming feeling of being lost, often leading students to think that they
are incapable of success in math. In most cases what is really wrong is
how they go about learning it. The most common errors students make in college
calculus involve poor application of the fundamentals of algebra, which then
only compound their errors derived from poor understanding of the fundamentals
of calculus.

I personally can contribute little to the mountains of
research on effective learning in mathematics. Yet, my experience as a college
professor, whose courses involve applications of mathematics, provides me some
insights on how students, parents, instructors, and education planners might improve
outcomes:

- Place more emphasis on understanding the fundamentals that motivate a rule.
- Do not memorize rules beyond the small handful that is most fundamental to a subject (For example, in trigonometry, memorize the Pythagorean theorem and the definitions of sine, cosine, and tangent in terms of the lengths of the sides of a right triangle, and memorize a few of the most fundamental values of these ratios).
- Know how to recreate the rule from the fundamentals.
*Practice*proofs and exercises as needed, playing with the math to consider different correct pathways to the same answer, making mistakes and learning from them along the way. This kind of “play” with math is what actually led to most of the proofs found in textbooks and in the math literature. Most students who ultimately succeed in college math do so because they paid a price of individual effort, sometimes fraught with frustration, which led to better fundamental understanding.- Students who do not understand a concept should not simply move on to the next one and forget about it, but should be given the freedom to work at it until proficiency is attained. Being slower than a group of other students during K-12 years should not carry a stigma: Students need to start from somewhere.
- Students need to learn to find their own mistakes by asking themselves whether their answers make sense, and they should confirm answers by working through exercises in different ways. If students are assigned exercise sets, the assignments should be sufficiently long to provide enough practice, but sufficiently short that students can check themselves and play with the math a little, and they should be able to move on without penalty, leaving assignments incomplete, if they demonstrate understanding.
- When a student makes a mistake, the mistake should provide a teaching and learning opportunity that can be used to drive better understanding of the fundamentals.

These rules might seem obvious on paper, but the whole
system of a one-size-fits-all curriculum makes their application
extraordinarily difficult in most public school environments. A truly successful
math education that leaves nearly no one far behind must treat each student as
an individual. Responsibility for success rests on the whole chain of those
involved, beginning with the students themselves, their teachers, their
parents, and their peers. Students must be allowed to take ownership of their
own learning. In order to allow such ownership to develop, the system must not
force each individual to move at exactly the same pace: The upcoming steps in a
learning plan must adjust to the individual abilities and needs of the student.
Those who do not yet understand a concept need to work at it personally, with assistance from a good mentor.

I invite discussion on these and alternative solutions, in
the comment section below.

You may contact the author at proundy at Albany dot edu

The opinions presented here are those of the author alone and do not necessarily reflect the perspectives of my employer or the State of New York.

The opinions presented here are those of the author alone and do not necessarily reflect the perspectives of my employer or the State of New York.