In order to recoup their enormous investments in AI, tech companies are inventing narratives that inflate the value of their product. Large Languange Models (LLMs) are perfectly suited to create “custom lesson plans” for students. Therefore tech companies are promoting the idea, backed by no evidence, that learning mathematics can be improved by…wait for it…AI generated custom lesson plans! Let’s suppose that you had the most amazing AI generated, customized piano lesson. You passively absorb that lesson plan so that you know exactly where to place your hands and what keys to press with which fingers according to what is on the sheet music. Are you suddenly are ready to play at Carnegie Hall? Of course not: passive lesson plans, no matter how great, are not sufficient to learn to play the piano. Learning anything, whether it be piano, writing, or mathematics, requires practice, which we prefer to call engagement. AI tutors are promoted on a self-serving, false premise. Immersive exercises, in contrast, promote the key to all learning: engagement. Learning is more than just the acquisition of knowledge (another misconception promoted by the tech industry). Leaning means internalizing the subject and developing instincts for how it can be used and how its boundaries can be pushed forward.

Dr. Plaut, who is not a programmer, read all the hype about the programming prowess of AI, and asked Gemini to code a basic app involving a classic related-rates problem from calculus: the sliding ladder. His request was an honest one, not intended to trick AI. After all, how great it would be to not have to rely on software engineers to code his designs! So he carefully explained that he needed an app demonstrating a ladder sliding down a wall. Nothing fancy—just a vertical bar for the wall, a horizontal one for the floor, and a bar representing the ladder leaning against the wall. The ladder could be made to slide up and down the wall by the user sliding a button below the floor. Within minutes, he had his software! It had the wall, the floor, the ladder and the slider. But instead of leaning against the wall, the ladder was hinged at the corner between the floor and the wall, and when the slider was moved, it flapped in the air like a left hand waving. The point here is not that Gemini made a mistake—humans do that too. But a third grader has enough instinctive understanding of the laws of physics to know that ladders don’t act like that! AI obviously “knows” that as well. Yet it still programmed that flapping ladder. That’s the basic problem with LLMs. They “know” a lot of stuff and can find it quickly, which is incredibly useful to a well-trained professional. But there is no evidence that they “understand” anything, and they often demonstrate (to trained professionals) that they don’t. The tech industry says just wait—it will get better! Maybe, but the safe route is for students to actually learn to understand the mathematics and science that they may be called upon to use in the future. Then they can use this amazing tool for what it is: a vast, easily searchable library of knowledge, and not try to rely on it to be what it is not: intelligent.

The distinction between an “answer” and a “solution” is often lost in the day-to-day pedagogy of mathematics courses. A solution is the process for solving a problem, and an answer is the result  of that process. Mathematics teachers or professors will likely agree that the most important learning objective of any mathematics class is that the student becomes proficient at solving mathematical problems. Yet our courses and the materials that we use are still oriented around answers, which often depend on the kind of rapid, flawless, low-level calculation that hasn’t been required of STEM professionals since the 1970s.  It’s common to see student solutions that are perfect except for a 5th-grade-level arithmetic mistake. If we only look at the answer, the student’s work is “wrong”. If the answer is all that matters in the evaluation of the student (e.g., in a multiple choice exam), then the fact that the student actually understood how to solve the problem is completely lost.

Let’s compare two approaches to Immersive Exercises (IEs). One approach is to find a solution online, then figure out how to adapt that solution to the input format of the IE, including the requirement of fractional input, and enter every problem-solving step in real time. This approach assumes that the online solution even matches up with a solution path embedded in the IE. Because IEs allow students to solve problems rapidly by avoiding manual arithmetic, instructors can assign far more repetitions of exercises. Students quickly learn that inputting the steps from internet solutions in real time is a comparably inefficient way to complete all those repetitions. They are highly motivated to take the second approach, which is to just learn the problem-solving steps so they can practice rapidly without accessing and inputting new solutions for every repetition.

Many “supplemental materials” for traditional math textbooks, including ebooks, send students to external software to do computations that are too complex to do with pencil and paper. The problem with this strategy is that the external software is a black box that hides the mathematical process the student is learning. It gives an answer, but the student learns little about where the answer came from. External software can also consume instructional time and add headaches for the instructor, and for what pedagogical purpose? In contrast, Immersive Exercises (IEs) are highly intuitive and have essentially no learning curve from a user standpoint, so that the student can focus on learning what is most important: solving problems.

If you’re an instructor, you and your students will be using Dr. Plaut’s text Interactive Linear Algebra together with software licensed by our partner company, The Math Atlas. The software consists of Immersive Exercises for each topic in the text. The text includes a QR code (paper copies) or links (e-book) to Immersive Exercises called “Math Pauses” (MPs). All MPs should be assigned for the material that is covered in the course, due shortly after the corresponding textbook material is introduced in the class.

Instructors obtain information for each student on their rate of completion of each Immersive Exercise. Students who make correct decisions will get the correct answer, and therefore it is the student’s progress that matters, not their answer. Instructors may also assign optional supplemental written exercises from the text, most of which concern applications or mathematical proofs. A special exam mode is available for most of the Immersive Exercises. Depending on the learning objectives of the course, instructors may give additional written test problems as part of a hybrid online/written exam. Data on students’ progress on the MPs, which is subject to Family Educational Rights and Privacy Act of 1974 (FERPA) regulations, is kept private and secure on The Math Atlas website.

Interactive Linear Algebra is an integrated set of materials, with Immersive Exercises linked directly from the text (e-book version). Rather than providing “interactive” widgets that demonstrate how to solve problems–which are simply a software implementation of traditional textbook examples–the Immersive Exercises  accompanying Interactive Linear Algebra actually replace both traditional examples and exercises. Students learn by solving problems rapidly within the Immersive Exercises, initially supported by hints, while the software handles routine, low-level calculations. Students can practice far more solutions, unencumbered by 5th-grade arithmetic that is incidental to, but not important for, the actual solution.

Despite the tech company hype, there is no evidence that AI “solves” any problems, as opposed to simply finding a solution to a very similar problem. STEM professionals, more than ever, will have to be problem-solvers. The work of good problem-solvers will be greatly enhanced by AI, which is great at finding existing problem-solving methods (but unable to actually “understand” what it finds). The STEM professional of the future will use AI to to explore what is known using AI, and synthesize it into solutions to novel problems at the cutting edge of science. For example, AI is great at writing bits of code, leading to the unemployment of people who are simply good at writing bits of code. But design of truly innovative software requires human oversight. Immersive Exercises allow students to develop mathematical instincts on basic problems that will benefit them when they tackle much more complex ones in the future.

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