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Retainium research · An interactive essay

Mathematics is a structured data problem.

Mathematics is unusually well structured. Concepts depend on each other in ways you can write down, and almost every question tests one small truth: a definition, a rule, a property, a procedure.

Treat that structure as data and two things follow. You can find where a student’s trouble really starts. And you can derive the practice to fix it, instead of writing it by hand.

  1. Part oneKnowledge is a graph, not a list
  2. Part twoQuestions are derived from truths
  3. Part threeLearning is a loop, not a line

October 202612 minute read

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Part one

Knowledge is a graph, not a list

Where should a student practise?

When a student gets stuck on limits, the problem often isn’t limits. It’s three steps back, in the fractions they never quite got.

Most learning software can’t see that, because it decides what comes next from a list. The first idea is to replace the list with a map of what depends on what, and to use that map to find where a struggle really starts.

01Arithmetic65%02Fractions28%03Exponents62%04Geometry48%05Algebra52%06Functions56%07Exp & Logs60%08Trig Functions82%09Limits14%10Derivatives57%

A first calculus course as a syllabus: eight topics, one after another.

01The shape of knowing

Courses are lists

Here is a first calculus course the way most learning software sees it: ten topics, one after another. Finish one, unlock the next.

The order is real, but it tells you what the author wrote first. It says almost nothing about what a student needs to know before something new will make sense.

Draw the dependencies instead, and the list falls apart. Limits, trig and exponentials all rest on functions, and functions rest on algebra. Trig also needs geometry. Algebra needs fractions and exponents.

Each concept is a node. Each arrow runs from a prerequisite to the concept that needs it. That is a knowledge graph.

Look at where the arrows meet. Derivatives sit on top of three branches: limits, trig, and exponentials and logs. A gap in any of them surfaces there.

A student who “can’t do derivatives” might be fine with limits and lost in trig, or fine with both and shaky on logs. The list can’t tell those students apart. The graph can.

02Zooming in

Inside a concept

A concept is too big to practise. “Algebra” is really four smaller ideas, called units, and units have prerequisites of their own.

Some come strictly in order. Others, in the same stage, can be learned side by side. And some lean on other concepts entirely: algebra borrows from fractions and exponents.

A student stuck on factoring $x^2 - 5x + 6$ isn’t stuck on algebra. They are stuck on one unit, and the graph knows which one comes before it.

03Diagnosis

Follow the struggle upstream

Now give the graph a student. Every answer they give updates a mastery score on the unit it tested, and each concept’s score is the average of its units.

Anything under 60% is red. This student’s weakest concept is easy to spot. The obvious move is to give them more practice on it.

The weakest concept is where a student notices the problem. It is rarely where the problem started.

So walk backwards. From the weakest concept, follow its prerequisites, and keep going only while they are weak too. The last weak concept the walk reaches is the likely root cause. If nothing upstream is weak, the gap really is in the concept itself.

Try a few more students. The walk is the same every time; only the answer moves.

04Mapping

Courses already know what they teach

A graph is only useful if it knows what a student has been taught. Every course is already an outline: modules, pages, review sessions. Tie each page to the units it introduces, and at any moment the system knows what has been covered and what hasn’t.

Pick a page below and the graph lights up what it teaches. Review pages draw on everything above them.

Arithmetic3 unitsFractions2 unitsExponentsGeometryAlgebraFunctionsExp & LogsTrig FunctionsLimitsDerivatives

Numbers & Fractions introduces Order of Operations (Arithmetic), Signed Numbers (Arithmetic), Percentages & Ratios (Arithmetic), Understanding Fractions (Fractions), Operations with Fractions (Fractions).

Pick a page. The graph lights up the concepts it teaches. Each page is tied to specific units, and that link is what connects a course to the graph.

That link, from page to unit, is what turns a course from a reading order into something the system can reason about. It is also where part one ends: with a target. For our student, that target is fractions.

Part two

Questions are derived from truths

What should they practise with?

Knowing where to practise only helps if there is something to practise with, and a lot of it. Hand-written question banks run out, and students memorise them.

The second idea stands on its own, with or without a graph. The thing worth writing down isn’t the question. It’s what the question tests: a unit of truth. There are four kinds.

  • Rules

    $x^a \cdot x^b = x^{a+b}$

    Hold for any numbers, so every value is a new question.

  • Definitions

    A number is even if it equals $2k$ for some integer $k$.

    Fix what a word means, so each part can be blanked or swapped.

  • Properties

    Multiplication is commutative: $ab = ba$. Subtraction isn’t.

    Apply only in some situations, so the question is whether one applies here.

  • Procedures

    Find the length, then the area, then the cost.

    Ordered steps, so each step can be asked on its own.

Write a truth once, mark where it can go wrong, and the questions follow: blanks, choices, true or false, worked steps. Every wrong option is a specific misreading of that truth, so a wrong answer says which one.

Where a truth comes from is a separate question from what kind it is. Definitions are stipulated; rules, properties and procedures can be given or proved. “The derivative of $\sin x$ is $\cos x$” is a derived rule: proved from the limit definition, then used like any other rule.

And one truth can be asked about in more than one way. “What is the derivative of sine?” asks a student to recall it, the way they would a definition. “Differentiate $\sin 3x$” asks them to apply it. Recall and application are different skills, so one truth yields both kinds of question.

05Truth · A rule

One rule, endless questions

A rule

The derivative of $x^n$ is $nx^{n-1}$
generates

  1. n = 7

    What is the derivative of $x^{7}$?
    Answer: $7x^{6}$

  2. n = 6

    What is the derivative of $x^{6}$?
    Answer: $6x^{5}$

  3. n = 5

    What is the derivative of $x^{5}$?
    Answer: $5x^{4}$

8 questionsfrom 1 source written by hand

A template fixes the rule and varies the numbers. The power rule says the derivative of $x^n$ is $nx^{n-1}$, so every value of $n$ is a fresh question with a known answer.

The wrong answers matter as much as the right one. Each distractor is a specific mistake: raising the exponent instead of lowering it, adding to the coefficient instead of multiplying, forgetting the exponent changes at all. When a student picks one, the choice says which mistake they made.

Rule
The derivative of $x^n$ is $nx^{n-1}$

Instance 1 of 8 · parameters n = 5

What is the derivative of $x^{5}$?

Choose an answer. Every wrong choice is a mistake students commonly make.

For the exponent product rule, the traps are multiplying the exponents instead of adding them, landing one off, or turning the exponent into a coefficient. Same idea, different mistakes.

06Truth · A definition

Every definition is a quiz waiting to happen

A definition

A number is even if it equals $2k$ for some integer $k$
generates

  1. True or false

    A number is even if it equals $2k$ for some integer $k$
    Answer: True

  2. Multiple choice

    A number is even if it equals $\underline{\qquad}$ for some integer $k$
    Answer: $2k$

  3. Fill in the blank

    A number is even if it equals $\underline{\qquad}$ for some integer $k$
    Answer: $2k$

12 questionsfrom 1 source written by hand

A rule holds for any numbers. A definition fixes what a word means, and every course is full of them. Take one: “A number is even if it equals $2k$ for some integer $k$.” There is nothing to vary, but there is still plenty to ask.

Mark the parts that carry the meaning (we call them regions) and that one definition turns into a handful of questions.

A number is even if it equals $2k$ for some integer $k$

2 marked regions

Multiple choice

2

Fill in the blank

2

True / false

8

Flashcards

10

12 questions and 10 flashcards from one definition.

Each region gives a fill-in-the-blank, a multiple-choice question, and a true version of the definition. Each distractor gives a false one, which a student has to catch. Shuffle them in with flashcards and you get the deck below. Try it.

Study item 1 of 22

Choose a study item

Fill in the blank

A number is even if it equals $\underline{\qquad}$ for some integer $k$

22 study items from one definition: 12 questions and 10 flashcards, interleaved. Blanks, choices and false versions are built from its marked regions.
One definition a teacher already wrote, a dozen ways to check that it stuck.

07Truth · Properties

A proof is a chain of truths

Properties, chained in a proof

Prove that if $n$ is even, then $n^2$ is even.
generates

  1. Justify step 3

    $4k^2 = 2(2k^2)$
    Answer: Factoring out common factor

  2. Justify step 2

    $n^2 = (2k)^2 = 4k^2$
    Answer: Power of a product: $(ab)^n = a^n b^n$

  3. Justify step 1

    Since $n$ is even, $n = 2k$ for some integer $k$
    Answer: Definition of even number

5 questionsfrom 1 source written by hand

A proof is a list of claims, and every claim rests on a truth: a definition (like the one above), a step of algebra (a rule), or a property that holds in this situation, like integers being closed under multiplication. Knowing which property applies is often the whole skill. Make the reasons explicit, and a student can check a proof one link at a time, and the system can ask about any single link.

Proof

The square of an even number is even

Prove that if $n$ is even, then $n^2$ is even.
  1. Step 1definition
    Since $n$ is even, $n = 2k$ for some integer $k$

    Because: Definition of even number

  2. Step 2rule
    $n^2 = (2k)^2 = 4k^2$

    Because: Power of a product: $(ab)^n = a^n b^n$

  3. Step 3rule
    $4k^2 = 2(2k^2)$

    Because: Factoring out common factor

  4. Step 4property
    Since $2k^2$ is an integer, $n^2 = 2m$ where $m = 2k^2$

    Because: Closure of integers under multiplication

  5. Step 5definition
    Therefore $n^2$ is even

    Because: Definition of even number (converse application)

Each step makes one claim for one reason, on purpose. The wrong justifications on offer are the properties students actually reach for: closure under addition instead of multiplication, the definition of odd instead of even.

08Truth · A procedure

A procedure, one step at a time

A procedure

Find the length from the width, then the area, then the bags of soil, then the cost.
generates

  1. Re-roll 3

    Width 10 m, length 2w + 6, 18 m² per bag at \$17.50
    Answer: $\$262.50$

  2. Re-roll 2

    Width 10 m, length 4w + 3, 17 m² per bag at \$13.00
    Answer: $\$338.00$

  3. Re-roll 1

    Width 10 m, length 4w + 1, 16 m² per bag at \$9.50
    Answer: $\$247.00$

40 questionsfrom 1 source written by hand

Some truths are procedures: do this, then this, then this. Word problems are where a procedure strings rules together. This one has four steps, and each answer feeds the next: a length, then an area, then a number of bags, then a cost.

Each step is tagged with the rule it uses, so a wrong answer points at a specific rule, not just at “word problems”.

Applied problem

Community Garden Planning

A town is planning a rectangular community garden. The garden's length is 1 more than 2 times its width.

  1. (a)If the width is 10 meters, what is the length?
  2. (b)Using your answer from (a), find the area of the garden in square meters.
  3. (c)Each bag of soil covers 20 square meters. How many bags must the town buy?
  4. (d)Each bag costs $9.00. What is the total cost?

Each answer feeds into the next step. The numbers are generated from a seed, so re-rolling produces a fresh problem of the same shape.

Re-roll it and the numbers change but the shape doesn’t: same steps, same rules, a new problem.

Part three

Learning is a loop, not a line

What happens when you have both?

Each idea is useful alone. A graph without generated questions can point a student at fractions, but someone still has to write the fraction practice. A generator without a graph can produce endless questions, but it can’t decide which ones a student needs.

09Try it

Close the loop

Here are both ideas running on the student from part one. The graph traces back to the root cause. A question is derived from a truth in that concept. Answer it, update the map, and watch where the trace goes next.

  1. 1 · The graph

    Pick the target

    Trace back from the weakest concept to the unit most likely causing it.

  2. 2 · The questions

    Write the practice

    Derive questions from the truths in that concept: its definitions, rules, properties and procedures.

  3. 3 · The questions

    Read the mistake

    A wrong answer is a specific distractor, and each distractor is a known error.

  4. 4 · The graph

    Update the map

    The answer moves that unit’s mastery, and the next trace starts from there.

Then round again. Each idea works alone; together they close the loop.

The student from part one, after 0 answers

Arithmetic65%Fractions28% · practisingExponents62%Geometry48%Algebra52%Functions56%Exp & Logs60%Trig Functions82%Limits14%Derivatives57%

Practising Fractions

From the rule: Add fractions over a common denominator

Add $\frac{1}{3} + \frac{1}{5}$

A right answer moves mastery up. A wrong one moves it down a little, and its distractor names the mistake. Each pass gives the next trace better evidence, until the gaps that caused the struggle are closed.

10Retainium

Where this goes

Retainium already helps people write and organise study material, and Prep turns notes into cheat sheets and quizzes. Both ideas would sit underneath that work: authors tag what they teach, the graph relates it, and the generators turn it into practice.

You can try what exists today in Prep or the educator workspace.

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