PMRI-001 — Primary Mathematics Research Institute Series
eduKateSG Learning Sciences & Education Research Programme
Framework Version: PM-LS v1.0
Status: Foundational Conceptual Framework
Research Domain: Primary Mathematics / Learning Sciences / Diagnostic Education
Jurisdictional Context: Singapore Primary Mathematics, Primary 1–6
Last Reviewed: August 2026
Start Here: https://edukatesg.com/how-mathematics-works/
Research Classification
Article Type: Conceptual framework + evidence synthesis + research agenda
This article is not:
a randomised controlled trial, a causal impact study, or proof that a specific eduKateSG intervention produces a specified population-level effect.
This article does:
define the Primary Mathematics learning-system model that eduKateSG will use to organise subsequent research questions, diagnostic studies, classroom observations, intervention designs and research-to-practice articles.
Quick Read
eduKateSG is beginning from a simple research problem:
Why can two children receive the same Mathematics mark, make the same wrong answer, or struggle with the same chapter—and yet require completely different interventions?
Our proposed answer is that Primary Mathematics should not be modelled only as:
topics → worksheets → test → score
It is more usefully investigated as a learning system.
That system includes:
prerequisite knowledge
↓
conceptual understanding
↓
retrieval
↓
connections between ideas
↓
mathematical language
↓
representation
↓
strategy selection
↓
execution
↓
transfer
↓
checking
↓
regulation
↓
performance
A visible error can originate anywhere inside that system.
Therefore:
Wrong Answer ≠ Diagnosis
and:
Score ≠ Complete Learner State
This does not mean marks are unimportant.
It means marks are outputs generated by a larger system.
The eduKateSG Primary Mathematics Research Programme will investigate that larger system.
The Core Research Hypothesis
Our starting hypothesis is:
Primary Mathematics performance is better understood when the learner is modelled as a dynamic system of mathematical capabilities and dependencies rather than as a single score or collection of completed syllabus chapters.
This hypothesis is deliberately testable.
If the framework does not improve our ability to:
- identify meaningful learning weaknesses;
- distinguish different causes of similar errors;
- select better interventions;
- predict which learning survives;
- reduce repeated failure;
- improve transfer;
- or explain performance changes;
then components of the framework should be modified, merged or discarded.
That is an important research principle.
A research framework should not be protected from failure.
It should become better because failure is allowed to change it.
The current IES Standards for Excellence in Education Research similarly require theories and conceptual frameworks to be clearly articulated, with hypotheses that are testable and falsifiable, and with results capable of refining, modifying or discarding the framework. (ies.ed.gov)
Evidence Labels Used by eduKateSG
Before building the model, we need to distinguish different levels of claim.
This is part of the institutional architecture.
eduKateSG will use five evidence labels.
E1 — Established External Evidence
A claim supported by a sufficiently strong external research base, established assessment framework or authoritative curriculum source.
Example:
Visual representations can support mathematical problem solving.
The IES What Works Clearinghouse grades 4–8 Mathematics practice guide gives strong-evidence recommendations for helping students monitor and reflect on problem solving and for teaching students how to use visual representations. (ies.ed.gov)
E2 — Supported Research Synthesis
A conclusion eduKateSG draws by connecting multiple established findings.
Example:
Retrieval, representation and problem selection should be investigated as separate capabilities rather than treated as one undifferentiated “Maths ability”.
This may be strongly motivated by the literature without itself having been directly validated as one complete eduKateSG construct.
E3 — eduKateSG Practice Observation
A recurring pattern observed through teaching.
Example:
Some students solve correctly when a worksheet is labelled “Fractions” but fail to select fraction reasoning when the same structure appears inside a mixed paper.
This can generate a useful hypothesis.
But unless studied systematically, it should remain a practice observation, not be presented as universal causal evidence.
E4 — Working Hypothesis
A proposed mechanism that needs testing.
Example:
A student’s difficulty with a current topic may sometimes be better reduced by repairing an earlier high-leverage dependency than by increasing practice of the visible topic.
This is testable.
We can examine whether the proposed earlier repair actually improves later performance.
E5 — eduKateSG Design Principle
A practical operating rule derived from evidence, theory and repeated observation.
Example:
Diagnose before increasing worksheet volume.
A design principle guides intervention.
It should still remain open to revision when evidence changes.
Why These Labels Matter
Without evidence labels, several very different statements can look identical on a webpage.
For example:
“Research proves…”
may accidentally mix:
- established findings;
- interpretation;
- tutoring experience;
- and speculation.
A genuine research culture should separate them.
The purpose is not to make every article sound cautious.
It is to make every claim inspectable.
The reader should be able to ask:
Where did this idea come from?
What evidence supports it?
Is this established?
Is this our model?
Is this still being tested?
That is how a research institution earns trust.
Not by declaring itself correct.
By showing how its knowledge is constructed.
Part 1 of 3
Why Model Primary Mathematics as a Learning System?
Research Question 1
What exactly are we measuring when we say that a child is “good” or “weak” at Mathematics?
A school examination gives us valuable evidence.
A student might score:
54%
or:
71%
or:
93%
But that single output compresses many different processes.
Consider two students who both receive 70%.
Student A
- concepts generally strong;
- mathematical language strong;
- reasoning strong;
- calculation errors frequent;
- checking weak;
- time management weak.
Student B
- arithmetic accurate;
- routine procedures stable;
- representation weak;
- transfer poor;
- unfamiliar problem solving weak.
Both produce:
70%
But the learning systems producing that 70% are very different.
If both receive identical intervention because they share the same mark, the intervention may be badly targeted.
This produces the first principle of the framework:
Performance is observable. Learner state must be inferred.
The Score as Compressed Information
A score is extremely useful.
It answers questions such as:
- how many marks were obtained;
- how performance compares against an assessment standard;
- whether achievement changed across assessments.
But it cannot by itself tell us exactly how the mark was generated.
We can express this as:
Learner State
↓
Task
↓
Strategy
↓
Execution
↓
Assessment Conditions
↓
Observed Response
↓
Score
The score sits at the end of the chain.
So eduKateSG proposes treating the score as a compressed signal rather than the complete state of the learner.
Reopening the Compression
If a child scores 64%, we can ask:
Knowledge
What did the learner actually know?
Retrieval
Which known information could not be accessed?
Language
Which mathematical wording was misunderstood?
Representation
Which questions could not be transformed into useful diagrams, models or symbols?
Routing
Which problems were connected to the wrong strategy?
Sequencing
Which multi-step solutions broke because the steps were organised incorrectly?
Execution
Where did arithmetic or procedural errors occur?
Transfer
Which knowledge worked only in familiar formats?
Calibration
Which unreasonable answers were accepted?
Regulation
Where did attention, rushing or time interfere?
Now:
64%
becomes:
a research surface
rather than simply:
a judgement
Research Question 2
Is Primary Mathematics best represented as a sequence of topics or as a dependency network?
For curriculum design, topics are necessary.
Singapore currently lists the 2021 Mathematics Syllabus for Primary 1–6, updated in October 2025, as the current Primary Mathematics syllabus in 2026. (Ministry of Education)
A syllabus needs an organised progression.
But the learner eventually needs something more interconnected than a chapter list.
Consider multiplication.
Multiplication connects to:
- division;
- factors;
- multiples;
- fractions;
- area;
- volume;
- percentage;
- ratio;
- rate;
- multi-step problems.
Fractions connect to:
- division;
- equivalence;
- decimals;
- percentages;
- ratio;
- measurement.
Place value connects to:
- whole numbers;
- written algorithms;
- decimals;
- money;
- measurement;
- estimation.
So while the curriculum can be presented sequentially, usable Mathematics becomes networked.
The Primary Mathematics Dependency Network
The simplest version might look like:
Number Sense
↓
Place Value
↓
Operations
↓
Multiplicative Structure
↓
Fractions
↓
Decimals
↓
Percentage
↓
Ratio / Proportional Relationships
↓
Integrated Problem Solving
But this straight line is still inadequate.
The real structure contains cross-links.
For example:
Multiplication
↔
Division
↔
Fractions
and:
Fractions
↔
Decimals
↔
Percentage
and:
Multiplication
↔
Area
↔
Volume
This gives us a better idea:
Primary Mathematics should be investigated as a dependency graph.
Some nodes contain mathematical knowledge.
Some edges describe relationships between knowledge.
Some routes are required for solving particular problems.
Node Failure and Edge Failure Are Different
Suppose a learner knows:
multiplication
and knows:
division
but does not recognise their inverse relationship.
The problem is not necessarily:
missing multiplication.
Nor:
missing division.
Both nodes may exist.
The connection between them may be weak.
This distinction produces two different diagnostic possibilities.
Missing Node
Knowledge itself is unavailable.
Broken Edge
Relevant pieces of knowledge exist but their relationship is not operational.
The intervention should differ.
That distinction will become formalised in PMRI-003: The Primary Mathematics Weak-Link Atlas.
The Visible Chapter Can Be Downstream of the Real Failure
Consider:
“My child is weak at percentage.”
Possible explanation:
percentage concept missing
But another possibility is:
fraction understanding weak
↓
decimal relationship weak
↓
percentage representation becomes unstable
Or:
multiplication/division retrieval slow
↓
percentage procedure consumes excessive mental capacity
↓
multi-step percentage problem fails
In both cases the visible failure appears in percentage.
But the earliest useful repair may lie elsewhere.
This gives us the Weak-Link Hypothesis:
Some current Mathematics failures are generated by earlier dependencies whose instability becomes visible only when later topics place sufficient load on them.
This is currently an eduKateSG working hypothesis.
It should be tested, not merely repeated.
Research Question 3
What does it mean for Mathematics to have been learnt?
Suppose the child answers ten fraction questions correctly immediately after instruction.
Has the concept been learnt?
Possibly.
But there are several stronger tests.
Immediate Performance
Can the learner solve now?
Delayed Retrieval
Can the learner solve after time has passed?
Representation Transfer
Can the learner recognise the same Mathematics in another representation?
Context Transfer
Can the learner solve when the story changes?
Mixed Selection
Can the learner choose the method when several competing methods are possible?
Independent Performance
Can the learner solve without tutor prompts?
These are different states.
The IES evidence-based guidance on organising learning recommends spacing learning over time, interleaving worked examples with problem-solving, connecting abstract and concrete representations, using active retrieval through quizzing, and returning to important content after delays. (ies.ed.gov)
That evidence helps motivate our decision to treat immediate correctness and durable learning as different measurements.
Learning Is Not the Same as Assisted Performance
Imagine:
Tutor explains
↓
Tutor gives hint
↓
Child completes
The final answer may be correct.
But we should ask:
What happens when the hint disappears?
This produces another state distinction.
Supported Performance
The child can succeed with scaffolding.
Independent Performance
The child can succeed without scaffolding.
Transfer Performance
The child can succeed independently after the task surface changes.
All three states matter.
They should not be collapsed into:
“The student can do it.”
The Receiver Principle
eduKateSG uses the term Receiver for the point where useful educational effect becomes real.
For Mathematics tuition, the receiver is not:
- the worksheet;
- the teacher;
- the lesson plan;
- or the parent’s perception that the explanation was excellent.
The effect becomes real when the learner can use the capability.
Therefore:
Teaching delivered ≠ Learning received
A useful research programme should measure the receiver.
Can the learner:
retrieve
represent
select
execute
transfer
with less external assistance?
That is where the intervention either succeeded or failed.
Learning Continuity
This leads to one of the central constructs in the eduKateSG framework:
Learning Continuity
Learning Continuity is the preservation, availability and reconnection of knowledge across time, topics, representations and contexts.
It is broader than remembering a formula.
Suppose a child learned multiplication in P2.
By P5, multiplication should still remain available inside:
- fractions;
- area;
- percentage;
- volume;
- multi-step problem solving.
If multiplication has to be substantially rebuilt whenever one of these topics appears, continuity is poor.
The child is repeatedly paying for old learning.
Learning Debt
We call accumulated unrepaired dependency weaknesses Learning Debt or, in this domain, Mathematical Debt.
The idea is straightforward.
A weak prerequisite imposes an additional cost whenever a later topic depends upon it.
For example:
weak multiplication
may increase difficulty in:
division
↓
fractions
↓
percentage
↓
ratio
↓
area / volume
↓
multi-step problem solving
The original weakness may be small.
Its downstream effects can expand.
The debt metaphor is currently an eduKateSG conceptual model, not a formally validated metric.
Future research should determine whether useful operational measures can be built around it.
Synchrony
Another construct is Learning Synchrony.
A learner is in reasonable synchrony when:
prerequisite capability
- ●
current instruction
- ●
practice demand
- ●
cognitive readiness
- ●
feedback
- ●
next-step difficulty
are sufficiently aligned.
Perfect alignment is neither possible nor desirable.
Learning requires challenge.
But large mismatches can create excessive friction.
Consider a P5 student learning percentage while basic fraction understanding remains unstable.
The learner now has two simultaneous jobs:
learn percentage
and:
repair fractions during percentage
That increases load.
When several such gaps accumulate:
Current Learning Demand
- ●
Repair Demand
can become larger than the learner can comfortably manage.
The Synchrony Hypothesis
eduKateSG therefore proposes:
Learning becomes less efficient when the repair load created by earlier unstable dependencies grows too large relative to the learner’s capacity to engage with current instruction.
Again, this should eventually be tested.
Possible observable indicators might include:
- increasing latency;
- greater tutor prompting;
- repeated prerequisite failures;
- declining transfer;
- rapidly forgotten procedures;
- growing discrepancy between guided and independent performance.
This moves “falling behind” away from a vague label.
It becomes a system-state question.
Part 2 of 3
The eduKateSG Primary Mathematics Learning System
We now need to define the proposed architecture.
The framework begins with twelve interacting components.
1. Prerequisite State
Research Question
What earlier Mathematics must be available for the present task to be learnable?
A prerequisite is not merely something taught in an earlier year.
It is knowledge or capability that the current learning process depends upon.
For example:
Percentage may depend on:
- multiplication;
- division;
- fraction understanding;
- decimal relationships;
- part-whole reasoning.
If those dependencies are unstable, the apparent percentage problem becomes more complex.
So the first research question for any difficult topic becomes:
What does this task assume the learner can already do?
2. Conceptual State
Research Question
Does the learner understand the mathematical relationship, or only reproduce a procedure?
These are different.
A child may know:
Average = Total ÷ Number of Items
without understanding average as an equalised value.
A child may know:
Area = Length × Breadth
without understanding why two dimensions are multiplied.
Conceptual state matters because unfamiliar questions often disturb surface procedures.
The deeper relationship is what allows reconstruction.
3. Retrieval State
Research Question
Can the relevant Mathematics be brought online when needed?
A learner may recognise a formula after seeing it.
That does not mean the formula was independently retrievable.
Retrieval is therefore a separate capability.
Evidence-based learning guidance supports spaced review and active retrieval as ways of strengthening longer-term access to previously learned material. (ies.ed.gov)
For the eduKateSG programme, the research question becomes more specific:
Which Mathematics requires near-automatic retrieval, and which Mathematics benefits more from reconstruction and reasoning?
Not everything should be memorised.
Not everything should be reconstructed from first principles every time.
The balance matters.
4. Connection State
Research Question
Are relevant mathematical ideas connected?
Examples:
multiplication ↔ division
fraction ↔ decimal ↔ percentage
length × breadth ↔ area
part ↔ whole
ratio ↔ scaling
A learner may possess the components separately.
The educational problem may be the connection.
This is why the framework distinguishes:
knowledge quantity
from:
knowledge topology
Two learners may know the same number of facts.
One may possess a far more useful network.
5. Translation State
Research Question
Can mathematical language be transformed into mathematical relationships?
Consider:
Mei has 7 more stickers than Ali.
The learner must identify:
- quantities;
- comparison;
- relationship;
- unknown.
This is not pure arithmetic.
Language is carrying mathematical structure.
The framework therefore treats mathematical reading as part of mathematical performance.
A learner can be computationally capable and still fail through translation.
6. Representation State
Research Question
Can the learner choose or construct a representation that makes the problem structure visible?
Possible representations include:
- objects;
- drawings;
- number lines;
- bar models;
- tables;
- graphs;
- symbolic statements;
- equations.
The IES Mathematics problem-solving practice guide gives strong-evidence support to teaching students how to use visual representations and also supports exposure to multiple strategies and explicit articulation of mathematical concepts. (ies.ed.gov)
This gives representation a central place in our framework.
It is not merely:
draw something.
It is:
transform the problem into a form that reduces cognitive difficulty and reveals the relevant relationship.
Representation as Cognitive Offloading
A complex problem may contain more information than a child can comfortably hold in working memory.
A representation moves part of that state into the environment.
Instead of mentally retaining:
quantity A
quantity B
relationship C
unknown D
the learner externalises them.
This gives us another hypothesis:
Useful representations may improve problem solving partly because they reduce the amount of problem state that must be maintained internally while making relationships easier to inspect.
The precise mechanism deserves careful empirical treatment.
For eduKateSG, the practical implication is simpler:
Representation choice should itself be observed and measured.
7. Routing State
Research Question
Can the learner choose which Mathematics to use?
This is one of the most important proposed constructs.
Consider a worksheet labelled:
Fractions
The learner has already received hidden help.
The worksheet has supplied the first strategic decision:
Use fraction Mathematics.
Now consider a mixed examination.
The student encounters:
- geometry;
- percentage;
- whole numbers;
- fractions;
- ratio;
- volume.
The method is no longer announced.
The learner must route the question into the appropriate mathematical system.
We call that:
Routing
Routing is the selection of a relevant mathematical structure, strategy or knowledge path from several plausible alternatives.
Why Routing Matters
A child can know five methods and still fail because the wrong one was chosen.
This is different from:
not knowing the method.
The IES problem-solving guide recommends exposing students to multiple problem-solving strategies. (ies.ed.gov)
But once multiple strategies exist, another capability becomes necessary:
choosing among them.
That is the routing problem.
8. Sequencing State
Research Question
Can the learner organise dependent mathematical steps?
A multi-step problem is not difficult merely because it has several calculations.
Often:
Step 2 requires Step 1
and:
Step 3 requires Step 2
The learner therefore needs to infer a dependency sequence.
This can be represented as:
Known State
↓
Find Intermediate A
↓
Use A to find B
↓
Use B to find target
This is planning.
A learner may know every individual operation and still fail the whole problem because the dependency sequence was not recognised.
9. Execution State
Research Question
Can the chosen Mathematics be performed accurately and efficiently?
Execution includes:
- arithmetic;
- algebraic-like manipulation at an age-appropriate level;
- unit control;
- formula use;
- written procedure;
- calculator use where applicable.
An execution error differs from a conceptual error.
This distinction matters greatly.
Suppose the child chooses the correct method and sets up the correct reasoning but calculates:
7 × 8 = 54
The intervention should not necessarily reteach the problem-solving concept.
The execution layer failed.
10. Transfer State
Research Question
Does the Mathematics survive a change in surface form?
A learner might succeed when:
numbers are the same
wording is similar
diagram looks familiar
chapter is announced
Transfer testing changes these.
Possible transfer levels:
Near Transfer
Different numbers.
Representation Transfer
Same relationship, different diagram or notation.
Context Transfer
Same Mathematics, new story.
Mixed Transfer
Competing topics present.
Farther Transfer
The learner has to combine known ideas in a novel way.
Transfer is central because classroom learning becomes educationally valuable when it remains usable beyond the exact training example.
Productive Difficulty and Transfer
Singapore-linked Mathematics education research on Productive Failure has investigated instructional designs in which learners first attempt novel problems before receiving subsequent instruction, with research examining how generation, failure and later consolidation can support learning. (repository.nie.edu.sg)
eduKateSG should not simply copy Productive Failure and relabel it.
Instead, it offers an important research lesson:
Early incorrect attempts can contain information about student thinking and can sometimes play a productive role when the subsequent instructional design uses them well.
This supports our decision not to treat every error as something that must be prevented immediately.
Some errors are useful sensors.
11. Calibration State
Research Question
Can the learner judge whether an answer is plausible?
Suppose a Primary student calculates:
The pencil is 400 metres long.
The arithmetic may have been executed.
But the answer should trigger suspicion.
Calibration includes:
- estimation;
- magnitude awareness;
- unit awareness;
- contextual reasonableness;
- self-checking.
A mathematically capable learner increasingly develops an internal model of what answers are plausible.
This is a form of self-monitoring.
12. Regulation State
Research Question
Can available mathematical capability survive the conditions under which it must be used?
The child may possess the Mathematics.
But performance may fail because of:
- rushing;
- attention;
- poor checking;
- excessive time on one question;
- inability to recover after difficulty.
This becomes especially visible during examinations.
So:
Capability
is not identical to:
Performance
Between them sits regulation.
The Full Proposed State Vector
The learner’s Primary Mathematics state can therefore be represented conceptually as:
M(t) =
Prerequisite State
- ●
Concept State
- ●
Retrieval State
- ●
Connection State
- ●
Translation State
- ●
Representation State
- ●
Routing State
- ●
Sequencing State
- ●
Execution State
- ●
Transfer State
- ●
Calibration State
- ●
Regulation State
The notation is conceptual.
We are not currently claiming that these components possess validated numerical coefficients that can be combined into a single scientific score.
That would require substantial measurement research.
At this stage, the value lies in distinguishing the constructs.
Do Not Create a Universal Maths Score
One possible temptation would be to create:
eduKateSG Mathematics Intelligence = 87.4
That would look scientific.
It might not be science.
A single number would immediately recreate the compression problem we are trying to solve.
Different dimensions may matter differently for different tasks.
Therefore the research programme should preserve a multidimensional state unless evidence later justifies stronger aggregation.
State Changes Over Time
The learner is also dynamic.
So:
M(t₁) ≠ M(t₂)
A child can:
- learn;
- forget;
- reconnect;
- become fluent;
- lose confidence;
- improve transfer;
- develop independence.
This means the purpose of diagnosis is not to permanently label the student.
It is to estimate the current state well enough to select the next useful intervention.
Student State Is Not Student Identity
This is an important ethical principle.
Saying:
“Fraction retrieval is unstable”
is very different from:
“This is a weak Maths child.”
The first statement describes a potentially repairable state.
The second converts current performance into identity.
Our framework rejects that move.
A diagnostic system should increase intervention precision.
It should not create permanent labels.
The eduKateSG Intervention Loop
Once the state is estimated, the intervention cycle becomes:
Sense
↓
Diagnose
↓
Prioritise
↓
Repair / Build
↓
Practise
↓
Retrieve
↓
Transfer
↓
Measure
↓
Update State
This is the Primary Mathematics control loop.
Sense
Observe:
- response;
- latency;
- method;
- representation;
- errors;
- explanation;
- independence.
Diagnose
Determine what mechanism most plausibly explains the difficulty.
Do not stop at:
wrong.
Ask:
Why wrong?
Prioritise
Not every weakness deserves equal attention.
Ask:
Which repair has the greatest downstream value?
This leads to the concept of the high-leverage weak link.
Repair / Build
If something is missing:
build it.
If it exists but is disconnected:
reconnect it.
If it is understood but not retrievable:
retrieve it.
If it is available but poorly routed:
train discrimination.
Different state.
Different intervention.
Practise
Practice is necessary.
But practice should correspond to the state being trained.
Blocked practice may help stabilise a new method.
Mixed practice may later train discrimination.
Transfer tasks test portability.
Timed tasks test performance under constraint.
The word practice therefore describes several different intervention types.
Retrieve
Return after a delay.
If the Mathematics disappears, immediate performance overestimated learning durability.
Evidence-based guidance supports delayed review and retrieval as important learning tools. (ies.ed.gov)
Transfer
Change the problem.
If the learning collapses, inspect whether the learner had memorised a surface pattern rather than constructed a usable relationship.
Measure Again
The intervention is not complete when teaching ends.
Ask:
What changed?
This returns us to the receiver.
Part 3 of 3
From Framework to Research Institution
The purpose of PMRI-001 is not merely to create another educational model.
It is to change how eduKateSG itself operates.
If eduKateSG is going to function publicly as a research organisation, it needs institutional behaviours.
Not just institutional language.
Research Institution Principle 1
Publish the Research Question
Each research article should state what is being investigated.
For example:
Does delayed retrieval provide a better indicator of durable fraction learning than immediate post-instruction accuracy?
That is stronger than:
“Retrieval is important.”
One is testable.
The other is a statement.
Research Institution Principle 2
State the Theory of Change
A research intervention should explain why it is expected to work.
For example:
Weak multiplication retrieval
↓
high cognitive load in fraction problem
↓
multi-step reasoning becomes unstable
Intervention:
strengthen multiplication retrieval
Expected mechanism:
reduce low-level processing demand
↓
increase capacity available for fraction reasoning
Expected observable outcome:
improved independent performance on fraction problems requiring multiplication
Now the mechanism is visible.
IES research standards similarly emphasise clearly articulated theory and hypotheses, intervention components, implementation, transparency and the ability to refine or discard theories when results do not support them. (ies.ed.gov)
Research Institution Principle 3
Publish Evidence Boundaries
Every article should say what it establishes and what it does not.
For PMRI-001:
What We Can Say
- Primary Mathematics can usefully be investigated as multiple interacting capabilities.
- Research evidence supports treating representation, reflection, strategy, retrieval and spacing as meaningful learning variables. (ies.ed.gov)
- Singapore’s current Assessment for Learning infrastructure explicitly uses diagnostic information to identify precise learning gaps and develop targeted interventions. (SEAB)
What We Cannot Yet Say
- The complete eduKateSG twelve-state model is a validated psychometric instrument.
- The proposed state categories have independently established causal effects.
- A specific weak-link diagnosis necessarily causes a specific score improvement.
- Findings from eduKateSG students automatically generalise to all Singapore students.
These limits are part of the paper.
Not an embarrassment added afterward.
Research Institution Principle 4
Separate Observation from Experiment
A tuition environment generates enormous observational information.
Tutors repeatedly see:
- errors;
- explanations;
- hesitation;
- misconceptions;
- transfer failure;
- recovery;
- performance change.
This is valuable.
But classroom observation is not automatically experimental evidence.
So eduKateSG should distinguish:
Practice Observation
from:
Systematic Observational Study
from:
Quasi-Experimental Study
from:
Randomised Experiment
from:
Evidence Synthesis
The publication should declare which type it is.
That alone will greatly improve epistemic discipline.
Research Institution Principle 5
Record Failure
A marketing organisation has an incentive to publish:
What worked.
A research organisation must also learn from:
What did not work.
Examples:
- repair appeared successful immediately but disappeared two weeks later;
- transfer failed;
- a representation helped one class but confused another;
- additional practice increased speed but not unfamiliar-problem performance;
- a proposed weak link turned out not to be the bottleneck.
These findings are extremely useful.
The IES research standards explicitly call for transparent reporting of positive, null and negative findings rather than shaping research around predetermined conclusions. (ies.ed.gov)
That principle should become part of eduKateSG’s research identity.
Research Institution Principle 6
Version the Framework
This article is:
PM-LS v1.0
Future evidence may change it.
For example:
PM-LS v1.1
could modify the Routing construct.
PM-LS v1.2
could merge two gap classes if they cannot be distinguished reliably.
PM-LS v2.0
could introduce validated measures.
Versioning creates intellectual memory.
We can inspect:
What did we believe?
What changed?
Why?
That is very different from silently rewriting old claims.
Research Institution Principle 7
Make Methods Inspectable
Where ethically and legally possible, research publications should disclose:
- research question;
- sampling logic;
- measures;
- intervention;
- duration;
- analysis;
- exclusions;
- uncertainty;
- limitations.
Current IES open-science standards emphasise preregistration, open methods, open access and appropriately protected data sharing for applicable research. (ies.ed.gov)
eduKateSG does not need to imitate every requirement of a US federal research agency.
But the underlying principle is valuable:
Someone outside the organisation should be able to understand how the conclusion was reached.
Research Institution Principle 8
Protect the Learner
Research involving children demands particular care.
The research objective never outranks the learner.
eduKateSG’s eventual internal-data programme should therefore build explicit governance around:
- consent;
- privacy;
- anonymisation;
- minimum necessary data;
- access controls;
- retention;
- research/publication separation.
Until that infrastructure is formalised, the safest research outputs are:
- external evidence syntheses;
- conceptual models;
- methods papers;
- carefully anonymised aggregate practice observations;
- research agendas.
The research institution should be built carefully rather than merely named quickly.
Research Institution Principle 9
Research Must Change Practice
A research paper that sits online without altering teaching has limited operational value.
The full eduKateSG loop should be:
Research Question
↓
Evidence Review
↓
Model
↓
Intervention Design
↓
Teaching
↓
Observation
↓
Measurement
↓
Revision
↓
Publication
↓
Improved Intervention
This is knowledge mobilisation.
Research returns to the classroom.
The classroom creates new research questions.
That closes the loop.
Singapore Is Already Moving Toward Diagnostic Mathematics
There is a particularly important local convergence.
In 2026, SEAB’s Assessment for Learning tools for Primary Mathematics are explicitly designed to diagnose learning gaps and provide teachers with qualitative performance feedback. The current system includes MathsCheckPlus for P2 and P4 and CATalytics topical assessments for P5 and P6. (SEAB)
SEAB states that participating schools can use the data to:
- identify students needing additional support;
- determine precise learning gaps;
- develop targeted intervention;
- and improve lesson design using the resulting information. (SEAB)
MathsCheckPlus goes further at P4 through an adaptive assessment, while its reporting is intended to inform teachers about mastery and support targeted remediation. SEAB also describes using P4 MathsCheckPlus with CATalytics to identify specific topic weaknesses, intervene, and then examine how students manage the topic post-intervention. (SEAB)
This is significant for our research programme.
It validates the importance of the diagnostic problem.
It does not validate every eduKateSG diagnostic category.
That distinction matters.
Where eduKateSG Can Contribute
SEAB operates at national assessment scale.
MOE operates curriculum and school systems.
NIE and universities conduct formal education research.
eduKateSG occupies a different position.
Our potential contribution lies in high-resolution learning observation inside a very small teaching environment.
A three-student Mathematics group can create an unusual observation surface.
The tutor may see:
- the exact step where hesitation begins;
- the representation chosen;
- when a student copies rather than understands;
- whether a hint changes behaviour;
- whether understanding survives after support is removed;
- whether the same error returns next week;
- whether transfer succeeds.
The small group does not automatically produce better research.
But it can produce high-resolution questions.
That is where the research programme can begin.
The eduKateSG Mathematics Research Instrument
The tutoring environment can gradually become an instrument for observing learning.
Not by turning every child into an experimental subject.
But by making teaching more systematically inspectable.
For example:
Before Intervention
Record:
student response
method
latency
error type
support required
During Intervention
Record:
representation
prompt level
number of reconstruction attempts
successful explanation
Immediate Post-Test
Record:
independent performance
Delayed Retrieval
Record:
retention after time
Transfer Test
Record:
performance after surface change
This creates a much richer learning trace than:
worksheet score increased from 6/10 to 9/10.
But Measurement Must Remain Proportionate
There is a danger in making education too instrumented.
The learner still needs to learn.
The tutor still needs to teach.
If data collection interrupts the intervention, the measurement system is defeating its purpose.
So the research design principle is:
Measure enough to distinguish important mechanisms, but not so much that measurement becomes the lesson.
This itself is a research question.
The Initial eduKateSG Primary Mathematics Research Programme
PMRI-001 establishes the framework.
The next five papers progressively operationalise it.
PMRI-002
How Do We Know What a Child Actually Understands in Mathematics?
The eduKateSG Mathematics State Estimator
Research focus:
How should learner state be inferred from multiple observations rather than one mark?
We will examine:
- accuracy;
- latency;
- explanation;
- representation;
- retrieval;
- transfer;
- prompting;
- error recurrence.
PMRI-003
The Primary Mathematics Weak-Link Atlas
Why the Same Wrong Answer Can Have Different Causes
Research focus:
Can recurring mathematical failures be classified into intervention-useful categories?
Candidate categories:
- Missing Node;
- Broken Edge;
- Weak Link;
- Wrong Edge;
- Routing;
- Translation;
- Transfer;
- Calibration;
- Regulation.
The goal will not be to defend the taxonomy.
The goal will be to determine whether it improves diagnosis.
PMRI-004
Mathematics Learning Continuity from Primary 1 to PSLE
What Must Survive for the Next Stage to Work?
Research focus:
Which earlier mathematical capabilities become high-leverage prerequisites for later learning?
This will formalise:
P1
↓
P2
↓
P3
↓
P4
↓
P5
↓
P6
↓
PSLE
as a dependency network rather than simply six yearly syllabuses.
PMRI-005
When Has a Mathematics Weakness Really Been Repaired?
Retrieval, Representation, Interleaving and Transfer
Research focus:
What evidence should be required before we conclude that learning repair has survived?
Candidate repair criteria:
Immediate correctness
↓
Independent correctness
↓
Delayed retrieval
↓
Representation change
↓
Mixed selection
↓
Transfer
This article will connect directly with the external research literature on durable learning and mathematical problem solving. (ies.ed.gov)
PMRI-006
eduKateSG Primary Mathematics Research Agenda 2026–2030
What We Know, What We Think, and What We Still Need to Test
This becomes the institutional research roadmap.
It should publish:
- open research questions;
- evidence classifications;
- planned studies;
- research limitations;
- measurement problems;
- unresolved hypotheses;
- failed hypotheses;
- future AI-supported diagnostic work;
- privacy/data-governance boundaries;
- replication goals.
This is where the programme becomes visibly ongoing.
From Research to Primary Mathematics Tuition
The tuition programme now sits under the research architecture.
That matters.
Instead of:
We run tuition, and here is some research supporting our tuition.
the structure becomes:
eduKateSG Learning Sciences & Education Research
↓
Primary Mathematics Research Programme
↓
Primary Mathematics Learning-System Framework
↓
Diagnostic Models
↓
Intervention Models
↓
Primary Mathematics Tuition
↓
Observed Learning
↓
New Research Questions
Tuition becomes one application environment.
Research becomes the governing intellectual layer.
What Changes for a Parent?
The parent should still receive a simple answer.
If a child is struggling, we do not begin with:
Do more worksheets.
We begin with:
What is actually failing?
Then:
Is something missing?
Is it disconnected?
Is retrieval too slow?
Is the language misunderstood?
Is the representation poor?
Is the wrong method being selected?
Does the learning disappear after a week?
Can the child solve only with help?
That is practical.
The research framework exists so the practical answer becomes more precise.
What Changes for a Student?
The student no longer receives only:
wrong.
Instead, errors can become information.
The question changes from:
“Am I bad at Maths?”
to:
“Which part of this system needs work?”
That is a much more actionable problem.
A learner can repair:
- multiplication retrieval;
- fraction equivalence;
- representation;
- routing;
- checking.
Those are states.
They are not identities.
What Changes for a Tutor?
The tutor becomes less like:
a person who knows the answer.
And increasingly like:
an observer and designer of learning.
The tutor asks:
What happened?
↓
Why?
↓
What is the earliest useful failure?
↓
What intervention would distinguish between competing explanations?
↓
Did the intervention survive?
This is a research mindset.
What Changes for eduKateSG?
The organisation begins accumulating something more valuable than articles.
It accumulates a research memory.
Over time:
Question
↓
Hypothesis
↓
Test
↓
Failure
↓
Revision
↓
New Hypothesis
That creates cumulative knowledge.
The website becomes the public surface of that knowledge.
The eduKateSG Research Standard
For the Primary Mathematics Research Institute series, we propose the following publication standard.
Every research paper should include:
1. Research Question
What are we trying to understand?
2. Claim Type
External evidence, synthesis, practice observation, hypothesis or design principle?
3. Evidence Base
What research or data supports the argument?
4. Model
What mechanism is proposed?
5. Prediction
What should happen if the model is correct?
6. Failure Condition
What finding would weaken or falsify the model?
7. Practice Translation
What changes in teaching?
8. Measurement
How could the effect be detected?
9. Evidence Boundary
What cannot yet be concluded?
10. Version
What iteration of the model is being published?
This is our minimum research grammar.
A Scientific Site Should Publish Uncertainty
One final distinction matters.
A tuition website tends to optimise for certainty.
Best method.
Proven system.
Guaranteed improvement.
A research organisation should be comfortable publishing:
We know this.
Evidence suggests this.
We repeatedly observe this.
We hypothesise this.
We do not yet know this.
This result contradicted our model.
That is not weakness.
That is epistemic control.
Current Framework: What Is Established and What Is Proposed?
Reasonably Well Supported Externally
There is strong external justification for treating several processes as educationally important, including:
- visual representation;
- monitoring and reflection during mathematical problem solving;
- multiple problem-solving approaches;
- articulation of mathematical concepts;
- spacing;
- delayed review;
- retrieval;
- connections between abstract and concrete representations. (ies.ed.gov)
Singapore’s current assessment system also explicitly supports diagnosis of specific learning gaps followed by targeted intervention and subsequent monitoring. (SEAB)
eduKateSG Synthesis
We propose that these and other capabilities can be organised into one broader Primary Mathematics learning-system framework.
Still Requiring Validation
The following remain open:
- whether the twelve proposed state dimensions are optimally separated;
- whether some should be merged;
- how reliably tutors can identify each state;
- which diagnostic indicators are most predictive;
- how quickly each type of weakness can be repaired;
- whether weak-link repair produces predictable downstream improvements;
- how intervention effects vary between learners;
- which measures best capture durable transfer.
These are research questions.
That is precisely why the programme exists.
The Founding Model
The complete PMRI-001 architecture is:
Reality
What can the learner actually do?
↓
Sense
Observe mathematical behaviour.
↓
State Estimate
Infer:
prerequisites
concepts
retrieval
connections
translation
representation
routing
sequencing
execution
transfer
calibration
regulation
↓
Weak-Link Search
Where is the earliest useful failure?
↓
Priority
Which intervention creates the greatest expected downstream value?
↓
Intervention
Build / repair / reconnect / retrieve / practise.
↓
Receiver Test
Can the learner use it independently?
↓
Delayed Test
Does it survive time?
↓
Transfer Test
Does it survive surface change?
↓
Performance Test
Does it survive mixed and examination conditions?
↓
Update
What changed?
↓
Research Memory
What does this case teach us?
↓
Model Revision
What should change in the framework?
↓
Publication
Make the knowledge inspectable.
↓
Next Research Question
Continue.
Conclusion
Primary Mathematics Is More Than a Sequence of Chapters
Primary Mathematics begins with number.
But eventually the learner has to build a much larger system.
Numbers must remain meaningful.
Operations must become available.
Concepts must connect.
Old learning must remain retrievable.
Language must become Mathematics.
Problems must be represented.
Strategies must be selected.
Steps must be sequenced.
Calculations must be executed.
Learning must transfer.
Answers must be checked.
Performance must survive constraint.
When something fails, the visible wrong answer is only the final output.
The scientific question is:
What generated it?
That is where eduKateSG’s Primary Mathematics Research Programme begins.
Not with:
What worksheet should the student do next?
But with:
What is the state of the learning system?
Then:
Which part is constraining the learner?
Then:
What intervention should change that state?
Then:
How will we know whether the change survived?
Then:
What would prove our explanation wrong?
That final question is important.
Because the objective is not to construct a complicated vocabulary around tuition.
The objective is to become increasingly accurate about how children learn Mathematics.
If a model helps:
keep it.
If evidence improves it:
upgrade it.
If evidence contradicts it:
change it.
If a distinction cannot be measured:
question it.
If an intervention does not transfer:
do not call the repair complete.
That is the institutional direction.
eduKateSG should not become a research institution by writing:
We are a research institution.
It should become one by repeatedly demonstrating:
Question
↓
Evidence
↓
Model
↓
Test
↓
Failure
↓
Learning
↓
Revision
↓
Publication
And Primary Mathematics is where we begin building that system.
Research Status at Publication
PMRI-001: Foundational framework established.
Next Research Paper:
PMRI-002 — How Do We Know What a Child Actually Understands in Mathematics?
The eduKateSG Mathematics State Estimator
Primary research question:
What observations are required to distinguish what a student can recognise, retrieve, explain, represent, route, transfer and perform independently?
That is the next layer.