ne of the coolest things currently offered on the Art of Problem Solving website is our free adaptive learning system, Alcumus.

We created it over 10 years ago, because we wanted to learn more about how students learn.

At AoPS, we’re always refining our courses and materials—and ten years ago, our founders were especially curious about how students generally interact with mathematics. What’s the best way to get students to practice tackling challenging material? What material do they struggle with? How much do students improve as they practice different topics?

What better way to answer these questions than by using machine learning to learn about learning??

Alcumus tried to use machine learning, like this robot, to get better.

The first version of Alcumus came from some lofty goals. Back then, we thought it was going to be a learning machine that provided kids with problems, letting them learn exactly what they needed, exactly when they needed it. But we soon realized our initial method for doing this was not actually what works best for learners.

In fact, the most important thing we learned with our early version of Alcumus was about how students interacted with the system overall. What we learned allowed us to build the current system, a better Alcumus, where more than 120,000 students have answered 35 million problems and counting.

How did Alcumus get so big? Read on!

First, why call it “Alcumus”?

When brainstorming name ideas, Richard Rusczyk and the other AoPS founders wanted a name that had some meaning, but also started with the letter A, so it’d naturally show up at the top of lists. This mattered back in the ancient days of Google’s infancy, when being at the top of an alphabetized Yahoo list was the best way to show how important you were.

Perhaps not-so-obviously, Alcumus was the result. Alcumus, sometimes also spelled ‘Alkimos’ or ‘Alcimus,’ was the father of Mentor, the character in Homer’s Odyssey who looks out for Odysseus’s son and guides him as he grows up. (Yup, Mentor in the Odyssey is where the word “mentor” comes from.) With this fancy pedigree, Alcumus the Learning Tool was born.

Why create Alcumus in the first place?

AoPS has always been committed to providing the absolute best education for students. We were already good at writing textbooks, teaching online classes, and creating interesting homework problems, and we were hoping that Alcumus would provide insights about how to make even better tools for learning and practice in the future.

We thought Alcumus was amazing, and we were really excited about what we (and our students) would learn from it. We had collected a lot of problems, and we started with the system designed to adjust as it learned more about the students.

The very first thing we realized was that this scattershot approach was a truly terrible way to try and teach actual humans.  

Those first students who used the first version of Alcumus hated feeling like they were adrift in a sea of math. We weren’t giving them the guidance they needed to solve challenging problems. Even after they were able to answer two or three easier problems in a row, we needed to show them how to move to harder ones.

Instead of continuing to answer questions, many of these first students simply gave up using Early Alcumus.

The truth is…it’s a rare middle- or high-school student who’s willing to play a random system long enough that the machine learning environment can begin to select less random, more tailored questions for them.

Today’s Alcumus is much better than the early version—for reasons you might not expect.

The reason Early Alcumus didn’t work as well as we’d hoped was an interesting problem as a result of the difference in how machine learning works compared to how humans understand things.

Intuitively, you probably know that math works by building one concept on top of another. If you don’t know what the distributive property is, for example, you’re going to struggle with factorization. You don’t need to try those problems to know that you’ll also struggle with the topics that come after that, too. Early Alcumus didn’t have that structure.

Instead, Early Alcumus treated all math topics as isolated islands, trying a little of everything until it discovered what students did or didn’t know. It takes a long time for a single person to provide enough data for a system like that to work properly!

On top of that, math doesn’t exist as topics you learn isolated from each other. Some problems only work after a student has been exposed to multiple concepts, and if all topics are treated as islands, there’s no way for a machine to know when to show those problems.

The solution to this was to create a map. Math is a complex, interconnected web of topics, but those topics still have an order to them. Students still master math topics in Alcumus by solving problems that adapt to their ability level by getting more challenging as they learn. But instead of starting with the entirety of the math ocean as an option, the Alcumus system explores abilities in logically related topics.

After students choose a topic, Alcumus gives them different problems based on their individual skills and abilities. Each problem has a clear and detailed solution that can include links to readings in our textbooks, or videos where Richard or an AoPS team member explains the necessary concept. For instance, you can watch videos on why we calculate the area of a circle the way we do, or how to find the least common multiple of a set of numbers.  

Alcumus is an extension of the Art of Problem Solving philosophy.

We’re here to teach bright students who have the potential to go on to solve some of tomorrow’s biggest problems. This means that a key part of our job is making sure they develop the skills and the internal motivation they’ll need to tackle those challenging problems.

Yes, math is one of the skills they’ll need.

But they also need curiosity and determination to work on hard problems, plus perseverance to keep trying after they’ve gotten a problem wrong. That’s why Alcumus allows students to submit incorrect answers before it’ll let them give up and view the solution to a problem. Even if they select a topic focus that shows them problems way beyond their current abilities, Alcumus still has them give it a try before it shows them the right answer. And if a student makes a mistake, Alcumus lets them try again.

You can see how this works by testing out Alcumus yourself; it’s free to use for everyone, not just AoPS students!

Think about your typical day and the work you do—how often do other people expect you to answer a particular question exactly once, and to give an answer that’s 100% right on the first try?

That’s part of why Alcumus lets students answer questions more than once if needed. After all, we’re asking students to solve hard problems—and since we’re really challenging them, we know they won’t get everything right on the first try.

Learning that mistakes are part of the process is one of the most important lessons that our students learn from Alcumus.

Developing persistence can be an especially vital lesson for bright learners, who may be seeing material that makes them struggle for the very first time. Yet many learning systems out there weren’t built to help students overcome unhealthy degrees of perfectionism. Even worse, some of these learning systems end up reinforcing those tendencies, by rewarding students based solely on their speed or accuracy. Alcumus, on the other hand, has an achievement that students can only earn by answering questions right on the second try.

At Art of Problem Solving, we expect that our students will experience some frustration while they learn. We want them to master hard things and we know that temporary frustration is an important part of the learning process. Alcumus is a natural extension of that pedagogical philosophy.

What’s today’s Alcumus like?

As we keep improving Alcumus, we’ve worked to make sure that our students like using it. It wasn’t enough to eliminate busywork. Because we want students to enjoy solving hard problems, we gamified Alcumus with badges and quests that make it fun (and a bit addictive!) to keep learning math.

One of the things we added that helped students master their math skills was a map of math topics. This was a major improvement from Early Alcumus! Intentionally determining where each topic fit into the big-picture sequence of subjects was a huge step toward improving the student experience.

As more and more students have used Alcumus, we’ve discovered again and again that all our students are unique. Some of them have gaps in their math knowledge that we weren’t expecting. By mapping all thirteen thousand problems (and counting!) on Alcumus to different curriculum areas, we’ve helped students see how the math they’re learning in one area supports their understanding of another. After all, one of the things that makes teaching math so beautiful is how one topic or set of skills builds on another.  

That’s why there’s not one single, correct path for students to take through Alcumus. They don’t just start in Prealgebra and answer a certain number of questions before progressing through Algebra, Counting & Probability, and so on.

Alcumus is full of branches and options, depending on what a student does or doesn’t know. For example, there are many possible ways for students to learn how to solve a question about exponents. Alcumus lets them follow their curiosity and move from one problem to another, related problem. If they want to jump to a completely different topic, they can do that too!

It also gives them chances to learn new skills by showing them problems that are related to, but not exactly like, the ones they’ve already seen. Alcumus can also be set to a particular difficulty level, which means that student’s won’t see really, really hard problems unless they specifically ask for them or have sought out the challenge. They have to ask for them by setting the level to “Insanely Hard” (which is an actual difficulty setting)–or they excel THAT much as they progress through the system.

Who should play Alcumus?

Alcumus has problems suitable for students studying Prealgebra, Algebra, Geometry, Discrete Math, Probability, or Number Theory. Its problems are also specifically tailored as practice for the following AoPS courses and their corresponding textbooks:

  • Prealgebra 1
  • Prealgebra 2
  • Introduction to Algebra A
  • Introduction to Algebra B
  • Introduction to Counting & Probability
  • Introduction to Number Theory
  • Introduction to Geometry
  • Intermediate Algebra
  • Precalculus

You don’t have to be an AoPS student to use Alcumus, though—it’s entirely free, and even has dashboards that teachers and tutors can use to track their students’ progress in the system.

Alcumus can be a great supplement to regular math courses, in the same way that it pairs so nicely with our own courses. It also works particularly well as enrichment for students who don’t feel challenged by their regular coursework. Using a tool like Alcumus can also help parents test the waters on providing additional educational opportunities for their kids.

There almost as many ways to use Alcumus as there are Alcumus users, and we’re proud that all the work that’s gone into making it has made it the helpful tool it is today

Do you use Alcumus? Tell us—tweet @AoPSNews, or comment on our Facebook page!

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