MaplePrimes - Maple Learn Posts and Questions

Finding the Shortest Path Between Two Points

callumneily — Thu, 20 Aug 2026 19:57:07 Z

Defining Our Goal

Suppose you place two points on the x-y plane and ask a seemingly simple question: What is the shortest path connecting them?

The "obvious" answer is a straight line (the green curve), but how can we show that this is truly the optimal path? Imagine two fixed points (x₁,y₁) and (x₂,y₂) connected by some curve y(x).

To find the length of this curve, we zoom in on an infinitesimally small segment. If the curve changes by small amounts dx horizontally and dy vertically, then the Pythagorean theorem gives the tiny arclength ds:

Here, y' = dy/dx is the derivative/slope of y(x) with respect to x. The total length S of the curve, often referred to as the "action", is found by integrating these arclengths:

Notice that, unlike ordinary functions that take numbers as inputs, S[y] takes an entire curve y(x) as input and outputs a single number, S. Such a function is called a functional.

Our goal is to find the curve y(x) that minimizes the length S[y]. Instead of asking how the function (or "functional") S[y] changes for some change in x, we must ask how the total path length changes if we slightly deform the curve y(x) itself. This is the central idea behind a branch of calculus called "Calculus of Variations".

Digging Deeper

Let's predict that the shortest path between two curves is y(x). Without knowing that y(x) is a straight line (we haven't proved this yet!), our first prediction is probably incorrect. So, let's say the true ideal path is a nearby path y(x) + ε*η(x), where η(x) is the shape of the function that corrects our original prediction, and ε scales this correction function (i.e. controls how large the deformation is). To ensure that the endpoints remain the same, we require η(x₁) = η(x₂) = 0.

Substituting this new path into the expression for S[y] above, we get the path length S as a function of ε:

Notice that if ε = 0, we recover the original path y(x). If the original path does minimize S, then S has a minimum at ε = 0. That is,

So, to find the minimizing curve y(x), we find a curve y(x) such that dS/dε = 0 at ε = 0. Bringing the derivative inside of the integral for S[y] and using the chain rule, this means:

We can integrate this by parts using the substitutions:

Since we originally defined η(x₁) = η(x₂) = 0, the first term on the right disappears. We're left with:

Now, this must be true for any η(x) we choose, meaning:

Squaring both sides and rearranging to solve for y', we get that:

Hence, y(x) has a constant slope and is therefore a straight line. Note that there are multiple ways to prove that the shortest path between two points is a straight line (such as using the Triangle Inequality or algebraic arguments under Euclidean geometry), but this method of using least action has proven to have much deeper physical significance to more sophisticated questions, as discussed below.

Important Distinction

As many of you have probably noticed, this argument only holds in flat Euclidean space (the space most of us are used to, like the xyz plane). In curved spaces, the analogue of a straight line is called a geodesic.

The general idea is still the same, but the formula for distance changes depending on the geometry of the space.

For example, consider the surface of the Earth. If you want the shortest route from Toronto to London while remaining on Earth's surface, you don't follow what looks like a straight line on a flat map. You follow part of a "great circle".

On a sphere of radius R, for example, the infinitesimal distance is given using spherical coordinates:

So instead of minimizing the flat-space length

we minimize:

Applying the same calculus-of-variations machinery gives the geodesics of the sphere, which turn out to be great circles.

This becomes especially interesting in general relativity. Spacetime itself is curved, and free-falling objects follow geodesics through that curved spacetime. In that sense, Earth's orbit around the Sun can be thought of not as Earth being forced away from a straight path, but as Earth following the natural geodesic of curved spacetime.

Deeper Physical Significance

What we proved above is a remarkable fact, as the idea of minimizing the "action" functional is a fundamental characteristic of systems in nature. Instead of minimizing distance, physical systems obey the same mathematics to minimize quantities involving energy and time. For example, a hanging rope will form a shape called a catenary (a type of hyperbolic cosine function) to minimize its gravitational potential energy, not a parabola (which approximates the motion of projectiles) as one might think.

Similarly, light bends when moving between materials like air and water to minimize travel time.

So, never doubt the power of seemingly "simple" or intuitive mathematical results, as this elementary question of minimizing distance between two points uses the same mathematics that governs planetary motion, quantum fields, and spacetime itself!

Math Success in the Age of AI: Introducing the Maplesoft Math Success Platform

Karishma — Thu, 02 Jul 2026 19:06:37 Z

Last week, we launched the Maplesoft Math Success Platform.

This launch reflects a lot of conversations I’ve had over the past year with educators and institutions about what it means to teach and learn math in the age of AI.

Math Education in the Age of AI

At first, many of those conversations were about visibility. If students were completing homework, quizzes, and other assessments with help from AI, those results became harder to interpret. Did students understand the work, or had they copied down a solution that made sense in the moment without building the understanding needed to do something similar on their own?

That visibility still matters.

Over time, though, those conversations led to a more nuanced conclusion. The question is not simply how we prevent students from taking shortcuts. It is how we help them develop the mathematical judgment, intuition, and critical thinking they will need in a world where AI is part of how they learn and work.

In some ways, that has become even more important. When answers are easy to generate, students need to be able to test ideas, recognize when something does not make sense, explain their reasoning, and trust their own thinking.

The Maplesoft Math Success Platform

That is why I am proud to share the launch of the Maplesoft Math Success Platform.

Built on Maple, the platform brings together our math technology and extends it with analytics, AI-driven insights, targeted resources, and content expertise to help institutions support math learning in a more complete way.

It gives instructors and learning support teams better insight into where students are struggling, supports the creation of better questions and learning experiences, helps students move beyond the answer, and helps institutions respond to a world where AI is now part of how students practice, study, and get help.

You can learn more about the Maplesoft Math Success Platform on our website.

We also wrote more about the thinking behind this launch in our new whitepaper, Math Education in the Age of AI: From Grading Answers to Understanding Student Progress. It looks at why math education needs a new approach in the age of AI: one that helps instructors ask better questions, create learning experiences that build understanding, and use learning signals to see where students need support.

I’d love to hear what you think. How are you seeing AI change the way students learn, practice, and get help in math? And what kinds of tools or approaches do you think will be most important as math education continues to evolve?

The Staircase Paradox

callumneily — Sat, 28 Feb 2026 00:55:47 Z

In mathematics, us humans love to rely on intuition. It helps us make physical sense of phenomena and guide our thinking before formal reasoning is developed.

For example, approximating the derivative of a function at a point can be thought of intuitively as dividing the function’s rise by its run. As we shorten the distance we run, this ratio approaches the value of the function’s derivative at that point. See this in the demonstration from Maple Learn below.

It is impossible to fully grasp the idea of moving an infinitesimal distance, so we make it easier by asking: “If we move an extremely small distance to the right, how much do we move up?”.

Intuition is typically a beautiful tool for approximating limits, but limits tend to limit (pun intended) the utility of our intuition. A perfect example of this? The Staircase Paradox.

Consider any rectangle you’d like. In the following example, we'll use a rectangle of width 3 and length 4 for convenience, but this paradox extends to any rectangle.

The name of the game is to ask yourself: how far must we walk along the edge of the rectangle to get from the top left corner to the bottom right. Here, the distance is of course 7 units (3 units right and 4 units down). This looks like a bit of a scary fall, so let’s add some stairs.

Even with the stairs, we’re still travelling a total distance of 7 units (1.5 + 1.5 units right, 2 + 2 units down). To shorten the fall even more, we can keep adding more and more stairs.

The important thing to notice is that no matter how many stairs we add, the distance travelled is always 7.

Now you may be wondering, where exactly is the paradox? Well, imagine now we have an infinite number of stairs. Our intuition tells us that our path to the bottom becomes more like a slide instead of a staircase. The steps we take are infinitely small, so it seems like we’re just travelling in a straight line down to the bottom right corner. However, if this were the case, we would have a right triangle! Using the Pythagorean Theorem, the length of our travelled path would be sqrt(3²+4²) = 5.

In other words, our calculations from before were wrong! But... they can’t be wrong, because we saw that the total distance of 7 units travelled was independent of the number of stairs we added.

This is a consequence of something called the “Manhattan distance”, which is the distance you travel if you can only move horizontally and vertically, like navigating the grid of streets in Manhattan. No matter how small we make the steps in our staircase, we are still only moving right and down. We never actually move diagonally. So even though the staircase looks more and more like a straight line, its length is always computed using horizontal distance + vertical distance. The limit of the shapes is a diagonal line, but the limit of the lengths is not the length of that diagonal. And that’s where our intuition stumbles.

The key lesson of the Staircase Paradox is that a sequence of curves can converge to a straight line, while their lengths converge to something completely different.

This is one of the quiet but profound messages of higher mathematics: limits preserve some properties, but not all. Smoothness, shape, and position may converge nicely, while quantities like length, area, or curvature behave in more subtle ways. Mathematics has a gentle way of reminding us that how we measure something can matter just as much as what we’re measuring.

Viviani's Theorem

callumneily — Tue, 17 Feb 2026 06:06:38 Z

Some mathematical theorems don't just prove a statement to be true, but they reveal something beautiful that itches our intuition. Viviani's Theorem is one of those rare gems. The theorem says something profound but geometrically elegant:

For any point inside an equilateral triangle, the sum of the perpendicular distances from that point to the triangle's three sides is always equal to the height of the triangle.

Take the equilateral triangle below as an example:

By picking any point P, we can draw lines perpendicular to each edge of the triangle that touch point P. The length of these lines always add to 6, which we will discover is the height of this triangle.

To explore this more, former Maplesoft co-op student Michael Barnett made a Maple Learn document on Viviani's Theorem where this is seen in action, as shown below.

No matter where the point P moves, the sum of perpendicular distances from that point to the edges of the triangle always add to the same value: the height of the triangle.

To see why this is the case mathematically, consider the example from before:

More generally, if we let x, y, and z be the shortest distances from the point P inside the triangle to the sides AB, AC and BC, respectively, one can conclude that:

In words, the sum of these distances x+y+z is simply the height of the triangle, h, no matter where the point P lies inside the triangle.

Even after reading this proof, it may be tempting to think of cases where this would not be true. I like to think of Maple Learn as a playground for geometry, algebra and visualization to interact. It helps to convince us that even for many different cases (in fact, all cases!), this theorem holds true.

Viviani's Theorem is a reminder that mathematics isn't always about answers. Instead, it's about finding hidden harmonies that our intuition begs us to question and search for. With tools like Maple Learn, those harmonies buried in symbols and complicated definitions can be uncovered and explored.

If geometry ever felt too distant or abstract, this is your invitation to see it come alive!

Using Maple Learn in a Business Calculus Classroom: A Workshop Perspective

Asia Majeed — Tue, 10 Feb 2026 02:13:32 Z

While conducting a workshop for business calculus students one day, I was reminded of a familiar challenge. Many students approach calculus with hesitation, especially those in business programs who may not see themselves as “math people.” Even when they are following the steps, it is not always clear that real understanding is happening.

During the workshop, we were discussing inflation as an example of exponential growth. I wrote the model on the board and explained how prices increase over time. Students were taking notes, but their expressions suggested they were still trying to connect the formula to its meaning.

So I opened Maple Learn.

I entered the equation, and the graph appeared right beside it. Almost immediately, the mood in the room shifted. One student leaned forward and said, “Oh… that’s what inflation looks like over time.”

That simple moment captured why visualization matters so much in calculus.

One of the strengths of Maple Learn is how naturally it combines symbolic work and graphical representation in a single space. Students can write equations, perform calculations, and see the corresponding graphs without switching tools. This makes abstract ideas feel more concrete and easier to interpret.

Maple Learn also works well as a note-taking tool. During the workshop, students kept their formulas, graphs, and written explanations together in one organized document. Instead of passively copying, they were actively building understanding as they worked through the example.

What stood out most was how easily students began sharing their work. They compared graphs, discussed small differences in their models, and asked one another questions. The technology supported conversation and collaboration, helping create a sense of community rather than isolated problem-solving.

By the end of the workshop, students seemed more confident and engaged. The combination of visualization, structured note-taking, and peer sharing helped transform a challenging topic into something accessible and meaningful.

Experiences like this remind me that when students can see mathematics, talk about it, and learn together, calculus becomes far less intimidating and far more powerful.

Encouraging Mathematical Thinking in Calculus

Asia Majeed — Mon, 09 Feb 2026 05:43:53 Z

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:

What does this derivative actually represent?
How does the function behave when something changes?
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:

Specializing - exploring specific examples
Conjecturing - noticing patterns and testing ideas
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.