Despite decades of reform efforts, mathematics teaching in the U.S. has changed little in the last century. As a result, it seems, American students have been left behind, now ranking 40th in the world in math literacy.
Several state and national reform efforts have tried to improve things. The most recent Common Core standards had a great deal of promise with their focus on how to teach mathematics, but after several years, it seems changes in teaching practices have been minimal.
[quote position="left" is_quote="true"]Students are learning to follow a rote process to arrive at a solution.[/quote]
As an education researcher, I’ve observed teachers trying to implement reforms — often with limited success. They sometimes make changes that are more cosmetic than substantive (e.g., more student discussion and group activity), while failing to get at the heart of the matter: What does it truly mean to teach and learn mathematics?
Traditional mathematics teaching
Traditional middle or high school mathematics teaching in the U.S. typically follows this pattern: The teacher demonstrates a set of procedures that can be used to solve a particular kind of problem. A similar problem is then introduced for the class to solve together. Then, the students get a number of exercises to practice on their own.
For example, when students learn about the area of shapes, they’re given a set of formulas. They put numbers into the correct formula and compute a solution. More complex questions might give the students the area and have them work backward to find a missing dimension. Students will often learn a different set of formulas each day: perhaps squares and rectangles one day, triangles the next.
[quote position="full" is_quote="true"] Students become dependent on tricks and rules that don’t hold true in all situations.[/quote]
Students in these kinds of lessons are learning to follow a rote process to arrive at a solution. This kind of instruction is so common that it’s seldom even questioned. After all, within a particular lesson, it makes the math seem easier, and students who are successful at getting the correct answers find this kind of teaching satisfying.
But it turns out that teaching mathematics this way can actually hinder learning. Children can become dependent on tricks and rules that don’t hold true in all situations, making it harder to adapt their knowledge to new situations.
For example, in traditional teaching, children learn that they should distribute a number by multiplying across parentheses and will practice doing so with numerous examples. When they begin learning how to solve equations, they often have trouble realizing that this step is not always needed. To illustrate, take the equation 3(x + 5) = 30. Children will likely multiply the 3 across the parentheses to make 3x + 15 = 30. They might just as easily have divided both sides by 3 to make x + 5 = 10, but a child who learned the distribution method might have difficulty recognizing the alternate method — or even that both procedures are equally correct.
More than a right answer
A key missing ingredient in these traditional lessons is conceptual understanding.
Concepts are ideas, meaning, and relationships. It’s not just about knowing the procedure (like how to compute the area of a triangle) but also the significance behind the procedure (like what area means). How concepts and procedures are related is important as well, such as how the area of a triangle can be considered half the area of a rectangle and how that relationship can be seen in area formulas.
Teaching for conceptual understanding has several benefits. Less information has to be memorized, and students can translate their knowledge to new situations more easily. For example, understanding what area means and how areas of different shapes are related can help students understand the concept of volume better. And learning the relationship between area and volume can help students understand how to interpret what the volume means once it’s been calculated.
In short, building relationships between how to solve a problem and why it’s solved that way helps students use what they already know to solve new problems they face. Students with a truly conceptual understanding can see how methods emerged from multiple interconnected ideas; their relationship to the solution goes deeper than rote drilling.
[quote position="full" is_quote="true"]Many teachers feel pressure to teach to the test, drilling students so they can regurgitate information accurately.[/quote]
Teaching this way is a critical first step if students are to begin recognizing mathematics as meaningful. Conceptual understanding is a key ingredient to helping people think mathematically and use mathematics outside a classroom.
Conceptual understanding in mathematics has been recognized as important for over a century and widely discussed for decades. So why has it not been incorporated into the curriculum, and why does traditional teaching abound?
Learning conceptually can take longer and be more difficult than just presenting formulas. Teaching this way may require additional time commitments both in and out of the classroom. Students may have never been asked to think this way before.
There are systemic obstacles as well. A new teacher may face pressure from fellow teachers who teach in traditional ways. The culture of overtesting in the last two decades means that students face more pressure than ever to get answers correct on tests.
The results of these tests are also being tied to teacher evaluation systems. Many teachers feel pressure to teach to the test, drilling students so they can regurgitate information accurately.
If we really want to improve America’s mathematics education, we need to rethink both our education system and our teaching methods and perhaps consider how other countries approach mathematics instruction. Research has provided evidence that teaching conceptually has benefits not offered by traditional teaching. And students who learn conceptually typically do as well as or better on achievement tests.
Education expert Pasi Sahlberg is a former mathematics and physics teacher from Finland, which is known for its world-class education. He sums it up well:
“We prepare children to learn how to learn, not how to take a test.”
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Grieving couple comforting each other
This response to someone grieving a friend might be the best internet comment ever
When someone is hit with the sudden loss of a friend or loved one, words rarely feel like enough. Yet, more than a decade ago, a wise Redditor named GSnow shared thoughts so profound they still bring comfort to grieving hearts today.
Originally posted around 2011, the now-famous reply was rediscovered when Upvoted, an official Reddit publication, featured it again to remind everyone of its enduring truth. It began as a simple plea for help: “My friend just died. I don't know what to do.”
What followed was a piece of writing that many consider one of the internet’s best comments of all time. It remains shared across social media, grief forums, and personal messages to this day because its honesty and metaphor speak to the raw reality of loss and the slow, irregular path toward healing.
Below is GSnow’s full reply, unchanged, in all its gentle, wave-crashing beauty:
Why this advice still matters
Mental health professionals and grief counselors often describe bereavement in stages or phases, but GSnow’s “wave theory” gives an image more relatable for many. Rather than a linear process, grief surges and retreats—sometimes triggered by a song, a place, or a simple morning cup of coffee.
In recent years, this metaphor has found renewed relevance. Communities on Reddit, TikTok, and grief support groups frequently reshare it to help explain the unpredictable nature of mourning.
Many readers say this analogy helps them feel less alone, giving them permission to ride each wave of grief rather than fight it.
Finding comfort in shared wisdom
Since this comment first surfaced, countless people have posted their own stories underneath it, thanking GSnow and passing the words to others facing fresh heartbreak. It’s proof that sometimes, the internet can feel like a global support group—strangers linked by shared loss and hope.
For those searching for more support today, organizations like The Dougy Center, GriefShare, and local bereavement groups offer compassionate resources. If you or someone you know is struggling with intense grief, please reach out to mental health professionals who can help navigate these deep waters.
When grief comes crashing like the ocean, remember these words—and hang on. There is life between the waves.
This article originally appeared four years ago.