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Asia Majeed

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Encouraging Mathematical Thinking in Cal...

Asia Majeed 5 Maple Learn
derivative calculus computation
3 hours ago
1 0
 
As a calculus instructor, one thing I’ve noticed year after year is that students don’t struggle with calculus because they’re incapable.

They struggle because calculus is often introduced as a list of procedures rather than as a way of thinking.

In many first-year courses, students quickly become focused on rules: differentiate this, integrate that, memorize formulas, repeat steps. And while procedural fluency is certainly part of learning mathematics, I’ve found that this approach can sometimes come at the cost of deeper understanding.

Students begin to feel that calculus is something to survive, rather than something to make sense of.

Research supports this concern when calculus becomes overly mechanical; students often miss the conceptual meaning behind the mathematics. That realization has pushed me to reflect more carefully on what I want students to take away from my class.

Over time, I’ve become increasingly interested in teaching approaches that emphasize mathematical thinking, not just computation.
 
Thinking Beyond Formulas

When I teach calculus, I want students to ask questions that go beyond getting the right answer:
 
  1.  What does this derivative actually represent?
  2. How does the function behave when something changes?
  3. Why do certain patterns keep appearing again and again?

These kinds of questions are often where real learning begins.

In The Role of Maple Learn in Teaching and Learning Calculus Through Mathematical Thinking, mathematical thinking is described through three key processes:
 
  1. Specializing - exploring specific examples
  2. Conjecturing - noticing patterns and testing ideas
  3. Generalizing - extending those patterns into broader principles

This framework captures the kind of reasoning I hope students develop as they move through calculus.

What Helps Students See the Mathematics

One of the biggest challenges in teaching calculus is helping students see the mathematics, not just perform it.

It’s easy for students to get stuck in algebraic steps before they ever have the chance to build intuition. I’ve found that students learn more effectively when they can explore examples, visualize behavior, and experiment with ideas early on.

Sometimes that happens through discussion, sometimes through carefully chosen problems, and sometimes through interactive tools that allow students to test patterns quickly.

The goal isn’t to replace thinking it’s to support it.

A Meaningful Example

One activity highlighted in the study, Inflation and Time Travel, places exponential growth into a context students can relate to: wages and inflation.

When students adjust values, observe trends, and ask what happens over long periods of time, calculus becomes much more than an abstract requirement. It becomes a way of understanding real phenomena.

Activities like this remind students that mathematics is not just symbolic work on paper; it is a way of describing and interpreting the world.

Final Thoughts

For me, calculus is not meant to be a barrier course.

It’s meant to be a gateway into powerful ways of reasoning about change, structure, and patterns.

When students begin to specialize, make conjectures, and generalize ideas for themselves, they start to experience calculus as something meaningful, not just mechanical.

And as an instructor, that is exactly what I hope to cultivate in my classroom.
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