A tutor for people who teach themselves from a textbook
Studium is a tutoring system for self-study in subjects like math, physics and computer science. You bring a textbook, or just something you want to learn. Studium works through it with you one new idea at a time, and keeps a record of how each idea connects to what you already understood.
We call one of those steps a loop. Below, one loop is drawn out on a small example from calculus.
What you know so far, with lines where you already see a connection.
ln x is new, and not connected to anything yet.
Starting from area: ln x is the area under 1/t from 1 to x.
Comes easily: the slope of ln x is 1/x.
Starting from ex: ln x is the power of e that gives x.
Comes easily: ln(ab) = ln a + ln b.
Tutor, not counted: “Split the area from 1 to ab at a.”
You, counted: “The piece from a to ab is the piece from 1 to b, a times wider and a times shorter. Same area, so it’s ln b.”
What Studium keeps from this loop:
Started from
integral as area, 1/t, fundamental theorem
Route
ln x as the area under 1/t
Evidence
your explanation of why ln(ab) = ln a + ln b
Next, suggested
show this ln x is the inverse of ex
Illustration, not the real interface.
A small map of ideas. The learner already knows six: derivatives, the fundamental theorem of calculus, the integral as area, the curve 1 over t, the function e to the x, and inverse functions, with lines between the ones they already connect. A new idea, ln x, appears on its own. In the loop, lines are drawn to ln x from the integral as area, 1 over t and the fundamental theorem; from there, ln x is the area under 1 over t, and its slope 1 over x comes easily. For comparison, the picture then shows ln x reached from e to the x and inverse functions instead; from that side, ln of a b equals ln a plus ln b comes easily. Next, a separate check reads only the learner’s own explanation and ignores the tutor’s hint. Finally the loop is recorded: the starting points, the route, the evidence, and a suggested next loop, showing that this ln x is the inverse of e to the x.
Start from what you already know. Say you’re halfway through a calculus book. You know how a definite integral measures area. You know the curve 1/t, the fundamental theorem of calculus, derivatives, the function ex, and what an inverse function is. Those are the points in the picture, with lines where you already see how two of them relate.
The next idea in the book is the natural logarithm, ln x. You may have pressed the ln key on a calculator. For now it sits apart, connected to nothing you understand.
A loop starts from two or three things you know. Here they are area, the curve 1/t and the fundamental theorem. From those, ln x is the area under 1/t between 1 and x. The tutor explains where explaining helps, then gives you problems where you have to use the idea. Thinking those through is what finishes the understanding.
Learned this way, the fact that the slope of ln x is 1/x is almost immediate. Why ln(ab) = ln a + ln b takes a short argument.
The loop is closed once ln x is tied into those starting points and no longer sits beside them as one more fact to remember.
Suppose you had started somewhere else, from ex and inverse functions. Then ln x is the power you raise e to in order to get x. From this side, ln(ab) = ln a + ln b comes straight from the rules for exponents, and the slope being 1/x is the part that takes work.
Same ln x, understood differently. Most systems record a topic as known or not known. Studium records the starting points you came from, because they change what will be easy for you next.
You say when you think you’re done. A separate check decides whether the loop is closed. It reads what you said and wrote in your own words and asks whether ln x is really tied to the starting points. It doesn’t count how many problems you got right. It doesn’t treat the tutor’s explanation as evidence, however good the explanation was.
Most products let the tutoring model announce that you’ve got it. In Studium the one judging is kept apart from the one teaching.
Then Studium remembers the loop: where you started, the route you took, and what you said that showed you had it. A checkmark would lose all of that.
The next loop is planned from this record. You came to ln x through area, so a good next step is to show that your ln x is the inverse of ex. That ties together two parts of what you already know.
What using it looks like
Everything happens in conversation. Each project has one main conversation. A project is whatever you’re aiming at: a whole subject, one book, a single topic. Your textbook is material the project uses.
When something needs real work, the main conversation opens a separate conversation under itself. There’s one for talking through what you want out of the project, and one for each loop.
When a loop ends, the main conversation tells you how it went and offers a few options for the next loop. Each option comes with its reason, one is marked as the recommendation, and you pick.
Project: single-variable calculus
Main conversation
Talk: what you want out of this project
Loop: ln x, starting from area closed
That loop is closed. Options for the next one:
Show that ln x is the inverse of ex. Recommended. It ties together two things you already know.
Integrate 1/x for negative x. Short, and it finishes what you know about 1/x.
Logarithmic differentiation. It puts ln x to work on the problems at the end of the chapter.
Illustration, not the real interface.
How Studium differs from most products
It judges understanding
Most products measure progress by how many problems you got right, or by an estimated probability that you’ve mastered a skill.
We judge whether you understood: whether the new idea is connected to what you knew, going by what you said.
The judge is separate from the teacher
Most products let the tutoring model decide when you’ve got it.
We make that decision in a separate step, which trusts only what you produced.
It keeps where you started from
Most products store one bit per topic: known or not known.
We store the starting points and the route, because the same idea learned from different places is a different understanding.
Its value is across loops
One session with a strong model and a good textbook is already good, and we don’t claim to beat that on its own.
Our work is on what happens between sessions: keeping track of where you are, what you can do, where to go next, and how the pieces connect.
Two smaller differences. Your route grows out of what you know as you go, instead of following a fixed syllabus. And Studium is only for the learner: there are no accounts for teachers, parents or classes.
What we hold ourselves to
Every conclusion points back to the place in the book and the words you said that support it.
You can see what Studium has inferred about you, and you can object to it.
It doesn’t put labels on you.
It doesn’t make medical or psychological diagnoses.
Its judgments are made apart from one another, so one conclusion can’t talk the next one into agreeing.
Where the work stands
The design is essentially done. A prototype is being built. Nothing has been released, so there is nothing to download and nothing to sign up for. Studium is built on Claude models.
We haven’t shown that it works better than anything. Here is how we plan to check. A single loop should be no worse than a general chat model given the same textbook. Over many loops, it should be clearly better. That is the test the prototype is being built to pass.