What Must Survive for the Next Stage to Work?
PMRI-004 — Primary Mathematics Research Institute Series
eduKateSG Learning Sciences & Education Research Programme
Framework: Mathematics Continuity Map MCM v1.0
Status: Longitudinal Learning-Trajectory Framework
Research Domain: Primary Mathematics / Learning Progressions / Prior Knowledge / Transfer
Last Reviewed: August 2026
Research Classification
Article Type: Learning-trajectory framework + evidence synthesis + longitudinal research agenda
Primary Research Question:
Which earlier mathematical capabilities must remain sufficiently available for later Primary Mathematics to be learnt efficiently?
Secondary Research Question:
How should we represent P1→PSLE development without pretending that every child follows one fixed linear pathway?
Research Upgrade
The newest evidence produces an important correction to our earlier language.
We should not model Mathematics as one rigid chain:
A must always precede B must always precede C.
Research on learning trajectories instead treats progression as a research-informed model of how understanding typically develops, including important subgoals, prior knowledge and instructional activities. Learning trajectories are useful precisely because they help educators choose personalised next goals while building on what students already know. (Springer)
Therefore PMRI-004 replaces:
Prerequisite Chain
with:
Probabilistic Dependency Network
Some earlier capabilities are strongly load-bearing.
Some are helpful but not strictly necessary.
Some relationships depend on representation or instructional route.
Some students find alternative paths.
That makes the model both more accurate and more testable.
Quick Read
Primary Mathematics is cumulative.
But cumulative does not mean perfectly linear.
A learner may encounter:
number
then:
operations
then:
multiplication/division
then:
fractions
then:
decimals
then:
percentage
then:
ratio
then:
integrated PSLE problems
Yet later learning also loops backwards.
Fractions can strengthen division understanding.
Percentage can deepen fraction–decimal equivalence.
Problem solving can expose an earlier multiplication weakness.
So the eduKateSG Continuity model becomes:
Mathematics is a growing dependency network whose earlier structures must remain sufficiently available for later structures to operate.
The key word is:
sufficiently
Not every prerequisite must be perfect.
But if too many high-load dependencies become unstable:
current learning becomes increasingly expensive.
Learning Continuity
eduKateSG defines:
Mathematics Learning Continuity as the preservation, availability, reconnection and usable transfer of mathematical capability across time, topics, representations and contexts.
Continuity is stronger than:
“The student once learnt it.”
The real question is:
Can it still be used when a later problem calls for it?
Part 1 of 3
From Curriculum Sequence to Learning Trajectory
A Syllabus and a Learner Are Not the Same Object
A syllabus needs organisation.
The learner builds a network.
Singapore’s current Primary Mathematics syllabus provides a structured P1–P6 progression, but students arrive at each new stage carrying different levels of prior knowledge, fluency and conceptual connection. The current syllabus remains the 2021 Primary Mathematics syllabus, updated in October 2025. (ies.ed.gov)
The research problem is therefore:
How does the formal curriculum interact with the learner’s actual internal state?
Learning Trajectories
Recent work on formative assessment describes learning trajectories as cognitive models of how specific mathematical understanding typically develops. Such models can be used to identify prior knowledge, possible misconceptions and meaningful proficiency levels. (Springer)
Another 2025 study describes learning trajectories as research-based sequences of learning goals, tasks and scaffolds that can help teachers choose personalised next learning goals while monitoring student assets, not only deficits. (Springer)
This gives eduKateSG a better architecture.
The Continuity Map Has Nodes and Edges
Nodes
Mathematical capabilities.
Examples:
- place value;
- multiplication meaning;
- multiplication retrieval;
- fraction equivalence;
- decimal place value;
- percentage reference control.
Edges
Relations through which earlier capability supports later capability.
Examples:
multiplication → division
fraction equivalence → fraction operations
fraction/decimal relation → percentage
Not All Edges Are Equal
We propose five preliminary edge types.
D1 — Strong Dependency
Later learning repeatedly requires the earlier capability.
Example:
Multiplication fluency supports many upper-primary calculations.
D2 — Conceptual Dependency
The later concept is easier to understand when an earlier relationship is conceptually secure.
Example:
Fraction equivalence supporting percentage conversion.
D3 — Representational Bridge
Earlier and later ideas are connected through different representations of related structure.
Example:
fraction ↔ decimal ↔ percentage.
D4 — Performance Dependency
The later concept can be understood without high fluency, but examination execution becomes inefficient if the earlier skill remains slow.
Example:
complex fraction reasoning with weak multiplication retrieval.
D5 — Conditional Dependency
The relationship matters only for some problem types or instructional routes.
This category prevents overclaiming.
Dependency Is Not Destiny
If a child has weak multiplication:
later Mathematics becomes harder.
But not necessarily impossible.
Students compensate.
Tutors scaffold.
Alternative representations help.
Therefore:
weak prerequisite increases risk or learning cost
is usually more defensible than:
weak prerequisite makes later learning impossible.
This distinction is important.
The Continuity Hypothesis
eduKateSG proposes:
The more high-load dependencies that remain unavailable, slow or disconnected, the greater the repair demand carried into current learning.
That demand competes with the cognitive and instructional resources needed for the present topic.
Repair Demand
Suppose a P5 lesson introduces percentage.
Student A has:
- secure multiplication;
- stable fractions;
- stable decimals.
Student B has:
- slow multiplication;
- uncertain fraction equivalence;
- unstable decimal place value.
They receive the same percentage lesson.
But Student B is effectively learning:
percentage
- ●
fraction repair
- ●
decimal repair
- ●
operation compensation
at the same time.
The curriculum demand is identical.
The learner demand is not.
Synchrony
This gives us:
Learning Synchrony = sufficient alignment between available prerequisite capability, current instruction and next-stage demand.
Synchrony does not mean zero difficulty.
A learner should encounter challenge.
The concern is excessive simultaneous repair.
The Synchrony Load Model
Conceptually:
Current Learning Load
- ●
Repair Load
- ●
Execution Load
must remain within a range the learner can productively manage.
This is not a validated numerical equation.
It is a research model.
Possible indicators of poor synchrony include:
- unusually high prompting;
- repeated prerequisite reconstruction;
- slow progress despite effort;
- rapid forgetting;
- strong guided performance but poor independent work.
P1→P6 as a Continuity System
Now we can revisit the developmental runway.
Primary 1 — Number Construction
The learner begins formalising:
- quantity;
- number;
- place value;
- basic operations;
- mathematical language.
Continuity question:
Does number remain meaningful as operations become more complex?
Primary 2 — Operations Stability
The learner develops:
- larger place value;
- addition/subtraction reliability;
- multiplication/division foundations;
- early fractions.
Continuity question:
Can early number relationships become efficient enough to support P3 complexity?
Primary 3 — Structural Connection
The learner coordinates:
- complete multiplication tables;
- division;
- fractions;
- measurement;
- area/perimeter;
- multi-step problems.
Continuity question:
Can earlier operations now be selected and combined?
Primary 4 — Network Consolidation
The mathematical web becomes denser.
Continuity question:
Do P1–P3 structures remain retrievable while new fraction, decimal, factor/multiple and multi-step demands are added?
Primary 5 — Upper-Primary Integration
The system becomes more representation-dense.
Continuity question:
Can fractions, decimals, percentage, rate and spatial Mathematics operate together without excessive repair debt?
Primary 6 — Final Primary Integration
The final primary layer is added.
Continuity question:
Can the entire P1–P6 network become available under increasingly mixed and examination-like conditions?
PSLE — Performance Under Constraint
The curriculum becomes an output environment.
Continuity question:
Can six years of Mathematics remain dispatchable under time, uncertainty and mixed-topic demand?
Part 2 of 3
Mathematical Debt and Continuity Failure
Forgetting Is One Form of Continuity Failure
Suppose:
Fractions learnt in March
then:
not revisited until October.
If the learner has to reconstruct the concept almost from the beginning:
continuity was low.
This is not surprising.
Memory changes over time.
The research challenge is determining which Mathematics needs deliberate retrieval and at what intervals.
Disconnection Is Another Form
The learner remembers:
0.25
and remembers:
25%
but does not recognise their equivalence.
Nothing was fully forgotten.
The network connection is missing.
That is a continuity failure of topology rather than storage.
Routing Failure Can Also Break Continuity
The learner knows fractions.
But when fractions appear inside a ratio problem:
the relevant knowledge is not dispatched.
The knowledge exists.
The route into it fails.
Continuity therefore includes:
availability + connection + dispatchability
not memory alone.
Three Continuity Dimensions
C1 — Availability
Can the earlier capability still be accessed?
C2 — Connectivity
Can it connect to the current concept?
C3 — Dispatchability
Can the learner recognise when to activate it?
These become core PMRI-004 constructs.
Mathematical Debt
We can now refine Mathematical Debt.
Mathematical Debt is the additional future learning or performance cost created when important prior capabilities remain insufficiently available, connected or dispatchable.
This is a conceptual construct.
Not yet a validated metric.
Debt Can Compound
Example:
P2 multiplication instability
↓
P3 division friction
↓
P4 factors/fractions friction
↓
P5 percentage load
↓
P6 mixed-problem overload
The model predicts a possible propagation path.
It does not prove that every child follows it.
Longitudinal evidence would be required.
Important Upgrade: Repair Does Not Guarantee Automatic Spillover
A 2025 quasi-experimental Grade 5 intervention study is especially important for this framework. It targeted foundational arithmetic concepts and found improvement on the explicitly taught conceptual outcomes, but transfer to basic skills was not automatic. (ResearchGate)
This gives us a strong rule:
Do not infer downstream repair. Measure it.
If multiplication understanding improves:
test division.
If fraction equivalence improves:
test percentage.
If representation improves:
test transfer.
The dependency model must earn its edges empirically.
Dependency Validation
For an edge:
A → B
we should eventually test:
Prediction 1
Weak A should increase risk or difficulty on B.
Prediction 2
Improving A should sometimes improve B if A was genuinely constraining B.
Prediction 3
If A improves and B does not:
the edge may be weaker, conditional or misidentified.
This is how the Continuity Map can become scientific.
Direct vs Indirect Dependencies
Some edges may be direct.
Example:
multiplication fact retrieval → multiplication calculation.
Others may be indirect.
Example:
place value → decimal reasoning → percentage.
The framework should distinguish them.
Otherwise the graph will become a network where:
everything causes everything.
That is not useful.
Learning Trajectories Are Local Models
Research on design-based Mathematics education emphasises that theories developed around particular learning processes are often local to topics and educational levels and are refined through iterative empirical cycles. (Springer)
That gives eduKateSG another boundary:
We should build topic-specific continuity maps before claiming one universal Mathematics dependency graph.
Examples:
- Multiplication Continuity Map
- Fraction Continuity Map
- Percentage Continuity Map
- Problem-Solving Continuity Map
The global map can then be built from validated local maps.
The Fraction Continuity Example
A candidate fraction trajectory might include:
equal partitioning
↓
part-whole relationship
↓
fraction as number
↓
equivalence
↓
comparison
↓
operations
↓
fraction of quantity
↓
fraction–decimal relationship
↓
percentage
But this is a candidate learning trajectory.
Not a rigid staircase.
Research on fraction monitoring explicitly notes that learning-trajectory approaches are useful because complex outcomes depend on different areas of prior knowledge that must be integrated into later learning. (Springer)
Assets, Not Only Deficits
An important upgrade from recent learning-trajectory research is that trajectories help teachers identify what instruction can build upon, not only what students lack. (Springer)
This matters.
A child with a weak symbolic fraction procedure may have:
- strong visual reasoning;
- strong part-whole understanding;
- strong proportional intuition.
Those assets can become the repair route.
So continuity research should map:
available bridges
as well as:
missing dependencies.
Repair Path Selection
Suppose Student A learns best from:
visual fraction representation → symbols
Student B already has strong symbolic manipulation but weak conceptual meaning.
Their repair routes need not be identical.
The same destination may admit several trajectories.
This is why the Continuity Map should remain a network.
Part 3 of 3
Measuring Continuity
Snapshot Assessment Is Not Enough
A single assessment can tell us:
state now.
Continuity requires:
state across time.
Therefore PMRI-004 needs longitudinal observation.
Continuity Probe 1 — Delayed Retrieval
Can the learner access the concept after a meaningful delay?
Continuity Probe 2 — Cross-Topic Activation
Can earlier Mathematics be used inside a later topic?
Continuity Probe 3 — Representation Reconnection
Can the learner recognise the same structure after representation changes?
Continuity Probe 4 — Mixed Routing
Can the relevant earlier Mathematics be selected without a topic label?
Continuity Probe 5 — Repair Persistence
After intervention:
does the weakness stay repaired?
Continuity Probe 6 — Downstream Effect
After repairing A:
does B improve?
This is the key test of the dependency map.
Continuity Profile
A future profile might report:
Multiplication
Availability: Strong
Connectivity: Moderate
Dispatchability: Moderate
Fractions
Availability: Moderate
Connectivity: Weak
Dispatchability: Weak
Percentage
Current learning: Emerging
Now the tutor can see why percentage might be expensive.
The Continuity Index Problem
Again, we should resist creating:
Continuity Score = 78.4
until evidence justifies aggregation.
A profile is currently more scientifically honest.
Different dependencies matter differently for different tasks.
Repair Rate
The earlier eduKateSG runtime used:
Repair Rate ≥ Gap Formation + Forgetting
This remains useful as a design principle.
But it should not be presented as a measured law.
The empirical research question is:
Can we estimate whether meaningful weaknesses are being resolved faster than they accumulate?
Possible practical sensors include:
- number of recurring weak-link classes;
- reappearance rate after repair;
- percentage of prior concepts independently retrievable;
- amount of prerequisite reteaching required in current topics.
Synchrony Test
The learner may be losing synchrony when:
- current lessons increasingly require earlier repair;
- prompting rises;
- transfer falls;
- homework time expands substantially;
- old errors repeatedly reactivate.
The intervention goal is not necessarily:
go backwards completely.
Instead:
Current Track
- ●
Repair Track
operate together.
Dual-Track Repair
Example:
P5 school topic:
Percentage
Underlying weakness:
Fraction equivalence
Run:
Present Track
Keep the learner connected to percentage.
Repair Track
Strengthen fraction equivalence.
Then test:
Does percentage become easier?
This is a direct continuity experiment.
Research Programme
PMRI-004 proposes the following questions.
RQ1
Which earlier Primary Mathematics capabilities most strongly predict later learning?
RQ2
Which dependencies are conceptual versus performance-related?
RQ3
How much prerequisite instability can students compensate for?
RQ4
Which weak links produce the largest downstream learning costs?
RQ5
Which repairs produce genuine spillover?
RQ6
Which repairs remain local?
RQ7
Which learners use alternative successful trajectories?
RQ8
How should continuity be measured without excessive testing?
RQ9
Can continuity profiles predict future intervention need better than marks alone?
RQ10
Can early continuity repair reduce later PSLE repair debt?
These are the core longitudinal questions.
Evidence Boundary
Current research supports using learning trajectories as research-informed models for identifying prior knowledge, developmental subgoals and instructional next steps. (Springer)
Recent intervention evidence also supports the idea that foundational concepts can remain educationally important for older students and can be directly remediated, while simultaneously warning that transfer into untrained outcomes must not be assumed. (ResearchGate)
What remains unvalidated is the specific eduKateSG P1→PSLE Continuity Map.
That is the purpose of PMRI-004.
The Mathematics Continuity Runtime
Map Current Capability
↓
Identify Prior Dependencies
↓
Classify Edge Strength
Strong / Conceptual / Representational / Performance / Conditional
↓
Test Availability
Can the prerequisite be retrieved?
↓
Test Connectivity
Can it attach to the present concept?
↓
Test Dispatchability
Can the learner activate it independently?
↓
Estimate Repair Load
What past Mathematics is competing with current learning?
↓
Prioritise High-Leverage Repair
↓
Maintain Present Curriculum
↓
Repair Earlier Dependency
↓
Reconnect
↓
Test Current Topic Again
↓
Measure Spillover
↓
Delay
↓
Test Persistence
↓
Update Dependency Map
↓
Carry Forward
Conclusion
Mathematics does not reset every January.
Primary 1 remains inside Primary 2.
Primary 2 remains inside Primary 3.
Earlier number relationships remain inside fractions.
Fractions remain inside percentages.
Multiplication remains inside area, volume and ratio.
But this does not mean every learner follows one perfect line.
The better model is:
a growing, revisable network of mathematical capabilities
Some structures are highly load-bearing.
Some are conditional.
Some learners reach the same concept through different routes.
Some repairs spread.
Some repairs remain local.
That gives eduKateSG a much stronger research position.
We should not ask only:
What topic comes next?
We should ask:
What must remain available for that topic to become learnable?
Then:
Which dependency actually matters for this learner?
Then:
Does repairing it change the downstream state?
That final question protects the framework from assumption.
A dependency that predicts nothing:
should be weakened.
A supposed prerequisite that many learners bypass successfully:
should be reconsidered.
A repair that never transfers:
should not be credited with downstream effects.
A previously unknown pathway that repeatedly works:
should enter the map.
This is why the Mathematics Continuity Map is not a static curriculum chart.
It is an empirical model of how earlier learning survives—or fails to survive—inside later learning.
Research Status
PMRI-001: System
PMRI-002: Measurement
PMRI-003: Failure Taxonomy
PMRI-004: Longitudinal Continuity
Next:
PMRI-005 — When Has a Mathematics Weakness Really Been Repaired?
Primary research question:
What evidence is sufficient to distinguish temporary performance improvement from durable, independent and transferable mathematical learning?