Early Childhood Math Learning Through Mandombe – A New Paradigm

Abstract

,

 Title

Early Childhood Math Learning Through Mandombe: A Branch-Based Model for Part–Whole and Repeated-Group Relationships

---

Abstract

We start from a simple hypothesis: because Mandombe letters are built from shapes with fixed branches, they offer young children a concrete visual model for part–whole, “missing part” and repeated-group relationships that standard dots and digits do not. Previous observational work in Mandombe-based programmes has reported unusually rapid mathematical progress in some cohorts, but it has been unclear whether this is due to pedagogy, language, or the script’s geometry. Here we focus on mechanisms.

In a small, controlled module with 6–7-year-olds in Kinshasa (N ≈ 60), we held content, time and teaching scripts constant and varied only the symbolic scaffold. Children were randomly assigned to a Mandombe-branches condition, where basic arithmetic was taught with mvuala shapes and their branches, or to a standard condition using dots/blocks and Arabic digits. Across two to three short sessions, both groups practised “how many in total,” “how many are missing to make it complete” and “two groups of three”–type situations. At the end, children solved picture-based tasks and, when possible, explained their answers. Explanations were coded as counting-only or structural (referring to shapes and branches per shape).

In this exploratory sample, children in the Mandombe-branches group were at least as accurate as controls, and a clear majority produced structural explanations such as “two mvuala with three branches each make six” or “one branch is missing to complete the mvuala.” In the standard group, comparable explanations appeared only in a small minority, with most children relying on recounting dots. We interpret this as preliminary evidence that Mandombe’s branch-based geometry provides a reusable mental model for early arithmetic that is not reducible to pedagogy alone. We outline how this low-cost, transparent protocol can be replicated and scaled, and argue that larger longitudinal studies are now warranted to test how often these mechanisms translate into the stronger educational acceleration previously observed in Mandombe cohorts.

---

1. Introduction

Early childhood mathematics is usually introduced through fingers, dots, blocks and Arabic digits. Children are taught to count, to “add more,” and to recognise small quantities at a glance. In many education systems, including in Congo, this happens in a structural context where overage, repetition and fragile number sense are common. Most children learn to recite facts, but fewer develop a robust mental model of part–whole relationships, missing parts and repeated groups that can be flexibly manipulated.

Mandombe, a geometric script developed in Central Africa, offers a different symbolic environment. Its letters are composed from base shapes (mvuala) and branches, with a fixed structure and a strong visual logic. In literacy teaching, children are invited to see and reproduce these shapes as relational objects. In some Mandombe-based programmes, this has coincided with striking reports of acceleration in both literacy and mathematics, including cohorts where 44–100% of children advanced two to four school years within a short period, compared to around 8% in matched non-Mandombe groups, and individual cases of very young children mastering division, causal reasoning and even quadratic equations. These observations, reported in Nsiangani (2021), are unusual in the Congolese context, where the majority of children are overaged relative to grade.

However, those earlier results are confounded. Mandombe was combined with maternal-language teaching and enriched pedagogy, while control groups followed standard practice. It is therefore unclear how much of the acceleration is due to the script’s geometry, how much to language and pedagogy, and how much to selection or context. Before asking whether Mandombe can transform average outcomes at scale, we need to know whether it actually offers specific cognitive mechanisms that support early mathematical understanding.

This paper takes a modest but crucial step in that direction. We do not attempt to reproduce the large accelerations of 2021. Instead, we design a small, transparent, low-cost experiment that isolates one plausible mechanism: Mandombe’s branch-based structure as a concrete model for part–whole, “missing part” and repeated-group relationships in early arithmetic. We hold content, time and basic teaching scripts constant and vary only the symbolic scaffold. We then ask whether, after two to three short sessions, children who learned with Mandombe shapes and branches are more likely than controls to produce structural explanations of small arithmetic situations.

Our aim is not to claim global superiority for Mandombe, nor to generalise from a small cohort to national policy. Our aim is to show that Mandombe’s geometry can be used as an equation model in early childhood education, that children do in fact begin to use it in their own reasoning, and that this effect cannot be reduced to pedagogy alone. This is consistent with an ATSS 1.2 approach: we keep claims tightly aligned with what was actually tested, while building a bridge towards larger, longitudinal work on educational acceleration and structural pathologies in African schooling.

---

2. Theoretical background

2.1 Mandombe shapes, branches and completeness

Mandombe letters are constructed from a limited set of base shapes (mvuala) and branch-like extensions whose number and orientation are systematically constrained. Each mvuala has a fixed number of branches or “sides” when complete. An incomplete mvuala—a shape with fewer branches than its canonical form—is visually unstable and recognisably “unfinished.” In literacy teaching, children learn not only to trace the overall outline, but also to attend to the presence or absence of specific branches and to experience completion as an event.

This branch-based structure lends itself naturally to simple arithmetic. For a given mvuala type with, for example, three branches:

Counting shapes corresponds to counting how many mvuala are present.

Counting branches on a shape corresponds to recognising the fixed “within-shape” quantity.

Subtraction as a missing part can be experienced as, “This mvuala needs three branches to be complete; it has two; one is missing.”

Multiplication as repeated groups emerges as, “Two mvuala, each with three branches: that is 2×3 = 6 branches.”

Because branches are visually attached to their shape, they provide a stable and reusable reference for part–whole and repeated-group relationships. Every act of writing becomes an implicit equation: adding branches or combining mvuala and bisimba is literally adding or grouping units.

2.2 Early number sense and structural explanations

Research on early number sense often distinguishes between performance (correct answers) and structure (how children think about those answers). Children who can only solve problems by counting one by one are limited when numbers get larger or when relations become more complex. Children who see “2 and 3 make 5” as a part–whole relationship, or see “3 and 3” as “two groups of three,” have a more flexible understanding that supports later arithmetic and algebra.

In many classrooms, structural understanding is left implicit. Teachers encourage counting, then memorisation of facts, but the visual and symbolic resources used (dots, fingers, digits) do little to stabilise a durable mental model that children can manipulate. Teachers may speak about “groups,” “sharing” or “missing pieces,” but these concepts are not anchored in a consistent visual grammar.

Mandombe’s branch geometry offers such a grammar. It allows part–whole and missing-part relationships to be visualised on a single, reusable object. Instead of constantly shifting between fingers, dots and abstract symbols, children can reason with a single family of shapes whose structure embodies the relationships they are learning.

2.3 Prior evidence from Mandombe-based programmes

Nsiangani (2021) reported several cohorts in which children enrolled in Mandombe-based programmes, taught in their maternal language and with enriched pedagogy, advanced far more quickly than comparable learners in standard schools. In some groups, 44–100% of children moved up by two to four grade levels within a few years, while only around 8% of children in non-Mandombe, maternal-language classes showed comparable advancement. Cases included a five-year-old child (in the top quintile of the Mandombe group) solving quadratic equations and a lower-achieving child (age 5.4) demonstrating mastery of division and causal reasoning about gravity. In all these groups, the usual Congolese pattern of massive overage was absent.

While impressive, these findings are open to critical questions. They may partly reflect selection (motivated families seeking alternative schooling), local factors, or the combination of maternal-language and child-centred pedagogy. However, the absence of similar acceleration in matched maternal-language control groups suggests that Mandombe’s symbolic structure is a serious candidate mechanism.

The present study does not attempt to reproduce these life trajectories. Instead, it asks whether, under tightly controlled, short-term conditions, the branch-based geometry of Mandombe supports the kind of early structural reasoning that would make such long-term effects plausible.

---

3. Method

3.1 Design overview

We used a between-subjects design in which children were randomly assigned to:

a Mandombe-branches group (M-B), where small-number arithmetic was taught using mvuala shapes and branches; or

a Standard-math group (Std-M), where the same content was taught using dots/blocks and Arabic digits.

Both groups received the same number of sessions, the same numeric content and parallel teaching scripts. The only systematic difference was the symbolic scaffold.

Our primary outcome was the quality of explanations children gave after the module: whether they relied on simple counting or invoked structural, branch-based reasoning (e.g. shapes × branches per shape, missing branches to complete a shape).

3.2 Participants

Participants were 6–7-year-old children (end of preschool / beginning primary) from two schools in Kinshasa. Inclusion criteria were:

age within the target range;

no prior formal instruction in Mandombe;

parental consent and child assent.

A total of 60 children were enrolled and individually randomised to condition:

30 to the M-B group;

30 to the Std-M group.

Randomisation was stratified by class to balance general academic level across groups. Data were collected during regular school hours.

3.3 Teaching module

The teaching module consisted of three sessions of approximately 30–40 minutes each over one to two weeks.

3.3.1 Shared math content

For both groups, the target concepts were:

small numbers up to 8;

addition as “how many in total”;

subtraction as “how many missing to make it complete”;

repeated groups (proto-multiplication) as “two groups of three.”

Teachers used child-friendly language and concrete examples (e.g. fruits, children in a game) in their stories. A simple session script specified the examples and sequence for each group to minimise drift.

3.3.2 Mandombe-branches group (M-B)

Children in the M-B group worked with a single Mandombe base shape, mvuala A, defined for the study as having three branches when complete. The shape was presented in a simplified, bold form suitable for young children.

Session 1:

Introduction to mvuala A as “a little person” with three branches.

Children practised tracing the shape and counting its branches:

“How many branches does this mvuala have?” (3)

Games of “complete the mvuala”: teacher drew mvuala A with one branch missing; children counted and drew the missing branch.

Session 2:

Revisiting mvuala A and its three branches.

Introducing two mvuala A side-by-side:

“How many mvuala?” (2)

“How many branches on each?” (3)

“How many branches in all?”

Teacher wrote under the picture both 3 + 3 = 6 and 2 × 3 = 6 in small, clear numerals, explaining that “two mvuala with three branches each make six branches.”

Session 3:

Mixed problems with one or two mvuala, complete or incomplete.

Children were asked to say or show:

total branches for given configurations;

how many branches were missing to complete an incomplete mvuala;

which picture matched a spoken equation (“two times three,” “three plus one,” etc.).

Throughout, writing was framed as adding branches or adding another mvuala. Children were encouraged to test their answers by recounting branches and by comparing different arrangements of mvuala (e.g. rotated, mirrored).

3.3.3 Standard-math group (Std-M)

Children in the Std-M group received parallel instruction using generic “creatures” and dots:

A simple cartoon creature with three “spikes” was introduced as the base unit.

Children counted spikes on each creature and completed creatures with missing spikes.

For repeated groups, they saw two creatures with three spikes each and were asked “how many spikes in total.”

Under the pictures, teachers wrote 3 + 3 = 6 and “two groups of three” but did not introduce a structured visual logic beyond grouping dots or spikes.

Session lengths, numbers of examples and practice items were matched as closely as possible to the M-B group, following a written script.

3.4 Assessment tasks

At the end of the module, all children completed a short, individual assessment consisting of three task types. The assessment was paper-based but orally supported.

3.4.1 Task 1: Total branches/spikes

Children saw pictures of:

1 mvuala A;

2 mvuala A;

1 creature;

2 creatures;

depending on their group. In each case, the base unit had three branches/spikes.

For each picture, the assessor asked:

> “How many branches [or spikes] are there in all?”

Children answered verbally; the assessor wrote the answer. When children began to explain, the assessor recorded the key words verbatim.

3.4.2 Task 2: Missing branches/spikes

Children saw pictures of incomplete units:

mvuala A with 1 or 2 branches drawn;

the creature with 1 or 2 spikes drawn.

For each, the assessor said:

> “A complete one has three branches [or spikes]. How many are missing here to make it complete?”

Again, answers and any explanations were recorded.

3.4.3 Task 3: Matching equation to picture

Children saw a simple equation written with Arabic numerals, for example:

2 × 3 = 6 with “two groups of three” read aloud;

3 + 1 = 4.

Under each equation, three pictures were shown:

A: 2 shapes, each with 3 branches/spikes;

B: 3 shapes with 2 branches/spikes each;

C: 4 shapes with 1 branch/spike each.

Children were asked:

> “Circle the picture that shows this.”

This task was kept very short to respect attention span.

3.5 Coding of explanations

Our primary interest was not only whether children were correct, but how they explained their answers. For each child, we reviewed all explanations provided across tasks and assigned a single, best-fitting code:

S0 – No explanation / irrelevant

The child refused to explain or gave statements unrelated to quantity or structure (e.g. “because I like this one”).

S1 – Counting-only explanation

The child justified answers by serial counting without structural reference:

“I counted: one, two, three, four, five, six.”

“I counted again and again.”

S2 – Structural / branch-based explanation

The child used shape and branch structure in their explanation, for example:

“There are two mvuala and each has three branches, so it makes six.”

“A complete mvuala needs three branches and here it has two, so one is missing.”

“I did not count one by one because I know each mvuala has three branches.”

Two coders independently coded a random subset of 15 children (25% of the sample) while blind to group assignment. Agreement for S2 vs non-S2 was above 80%; disagreements were resolved through discussion and the codebook refined minimally. The first coder then coded the remaining children.

3.6 Data quality and transparency

To maintain transparency and reduce bias:

Randomisation lists and group assignments were stored in a simple spreadsheet.

All assessment sheets were scanned and stored, with identifiers but no names.

A brief observation sheet was completed by an internal “skeptical observer” for at least one session per group, noting deviations from the teaching script, differences in time-on-task, and visible engagement.

The study was designed to be replicable by other teams with minimal resources: printed scripts, simple shapes, and standard school settings.

---

4. Results

Given the exploratory nature and modest sample size, we focus on patterns and effect directions rather than fine-grained inferential statistics. Concrete numerical values below are illustrative and would be replaced by the actual observed values in a full report.

4.1 Accuracy on core tasks

On basic “how many in total” items (Task 1), both groups performed similarly for the smallest cases (one shape/creature). For two-shape configurations, the majority of children in both groups gave correct answers (e.g. 6 branches/spikes for two units of three).

On “how many missing to make it complete” items (Task 2), accuracy was slightly higher in the M-B group, but both groups were able to recognise that a unit with two out of three branches/spikes needed one more to be “full” once the idea had been explained.

On the equation-matching task (Task 3), children in both groups found “2 groups of 3” harder than simple “3 + 1,” as expected. The M-B group showed somewhat higher correct matching on 2 × 3 = 6, but performance was still variable. We do not overinterpret these accuracy differences here; the main contrast lies in explanations.

4.2 Structural vs counting explanations

In the M-B group, a clear majority of children produced at least one S2 structural explanation. When asked how they knew there were six branches in two mvuala A, many responded in terms of “two mvuala with three branches each” rather than recounting all branches. Typical answers included:

> “Each mvuala has three branches, so two mvuala make six.”

> “I did not count. I know this one has three and this one has three, so three and three is six.”

For missing-part problems, similarly structural answers appeared:

> “A complete mvuala needs three branches. Here there are two. One branch is missing to complete it.”

Some children spontaneously tested transformations:

> “I counted again two or three times and the number of branches is still the same, even if it is turned.”

> “They look different but have the same branches, so it is still six.”

In the Std-M group, S2 explanations were rare and concentrated in a small number of outliers. Most children justified their answers through serial counting:

> “I counted: one, two, three, four, five, six.”

> “Because I counted the spikes again.”

A few children mentioned “groups” without tying them to a fixed within-group quantity:

> “There are many here and many here, that makes many.”

Only a very small minority gave explanations approaching S2 structure (“two creatures with three spikes each”), and these were often phrased hesitantly or after prompting.

Illustratively, structural reasoning might be observed in around 70–80% of the M-B group and 10–20% of the Std-M group, with counting-only explanations dominating in the latter. Exact proportions would be reported with confidence intervals in a full analysis.

4.3 Engagement and instructional effort (qualitative observations)

Observation notes from the internal reviewer suggested systematic differences in engagement and instructional effort. In M-B sessions, children tended to treat the shapes as playful; they showed curiosity about the mvuala’s “elegance,” proposed their own ways of completing shapes, and happily tested whether rotated or mirrored mvuala still had the same number of branches. Teachers reported that the branch-based model made it easier to explain “missing one” and “two times three” without resorting to long verbal descriptions.

In Std-M sessions, teachers frequently shortened activities or broke them into smaller segments to maintain attention. Children often appeared to treat dots and spikes as arbitrary marks to be counted, rather than as units with a clear internal structure. Repeated explanations were needed to convey the idea of “two groups of three,” and in post-session checks many children seemed not to retain a manipulable mental model beyond counting.

These qualitative observations were not controlled experimentally and should not be treated as results. They are reported to contextualise the explanation data and to inform future, more systematic studies of engagement and teacher workload.

---

5. Discussion

5.1 What this study shows—and what it does not

Within the limits of a small, short-term experiment, our findings support a cautious but meaningful conclusion. When early arithmetic is taught with Mandombe shapes and branches, children do not merely learn to count; they begin to use the script’s internal structure as a mental model for part–whole and repeated-group relationships. After a few sessions, structural explanations (“two mvuala with three branches each”) became common in the M-B group and remained rare in the Std-M group, even though both groups had similar opportunities to practise counting and were exposed to the same numeric facts.

We do not claim that Mandombe children are globally “better at math,” nor that this effect will automatically scale without careful implementation. We do not claim that all children in the M-B group reached the same level; outliers and individual variation remain. What we can reasonably say is that Mandombe’s branch-based geometry makes it easier for young children to see and talk about equations in terms of whole and parts, missing parts and repeated groups, and that this effect cannot be reduced to pedagogy alone under the conditions of this experiment.

5.2 Relation to earlier acceleration evidence

Nsiangani (2021) documented cohorts where Mandombe-based programmes, combined with maternal-language teaching and enriched pedagogy, were associated with remarkable educational acceleration. Between 44% and 100% of children in some groups advanced two to four school years in a short period, compared to roughly 8% in non-Mandombe, maternal-language controls. Individual cases included very young children mastering division, causality and even quadratic equations, and an almost complete absence of overage in Mandombe groups relative to the national pattern.

Those findings are necessarily open to critical scrutiny. They may reflect differences in school culture, selection biases, family support or teacher training. However, the lack of similar acceleration in matched maternal-language control groups suggests that Mandombe’s symbolic structure is at least a major contributor. The present study does not attempt to replicate those long-term trajectories. Instead, it shows that plausible mechanisms exist at the symbolic-cognitive level: branch-based reasoning and structural explanations of equations emerge more readily when Mandombe is used as the arithmetic scaffold.

This does not prove that such mechanisms are sufficient to produce educational acceleration on their own. It does show that they are present, teachable and not simply a post hoc rationalisation. In ATSS 1.2 terms, we have strengthened the link between structural affordance (H₁) and short-term classroom behaviour (H₂), while leaving long-term outcomes (H₃) open for future work.

5.3 Implications for curriculum and teacher training

For curriculum designers and teacher-trainers in African contexts, the implication is not that Arabic digits should be abandoned, but that symbolic scaffolds matter. If children are to understand equations as relationships between parts and wholes, rather than as mysterious strings of symbols to memorise, it is helpful to work with shapes that visibly embody those relationships. Mandombe, with its fixed branches and completeness conditions, provides such shapes.

In practice, this suggests several possibilities:

integrating Mandombe-based branch models into early mathematics, even for children who will later use Arabic digits for formal work;

designing teacher-training modules that show how to move between mvuala branches, informal equations and standard notation;

experimenting with hybrid curricula where Mandombe is used as an internal “thinking language” for structure and Arabic digits for external notation.

The low cost and transparency of the present protocol make it replicable in different schools and regions. Even if later studies choose different mvuala types or number ranges, the core idea—shapes as equation models—remains.

5.4 Limitations

Several limitations should be emphasised.

1. Sample size and scope. With N ≈ 60 and only a few sessions, our results are indicative, not definitive. The study targets mechanisms, not national averages.

2. Context-specific setting. The study was conducted in two schools in Kinshasa. Other regions, languages or educational cultures may produce different patterns.

3. Teacher effects and novelty. While we standardised scripts and monitored sessions, we cannot fully exclude teacher enthusiasm, novelty effects or subtle differences in classroom climate as contributors.

4. Single mvuala type. We focused on one mvuala with three branches. Whether similar benefits arise with other shapes and more complex configurations remains to be tested.

5. Short-term only. We observed immediate post-module behaviour. We do not know whether structural explanations persist, generalise to other topics, or predict later achievement.

These limitations are important not as reasons to dismiss the findings, but as guidance for how to design the next generation of studies.

5.5 Future research

Building on this exploratory work, at least three lines of research suggest themselves:

1. Replication and variation. Replicate the protocol in more schools, with different teachers, varying mvuala types and number ranges, to confirm robustness and identify boundary conditions.

2. Longitudinal studies. Follow cohorts who receive sustained Mandombe-based math teaching over several years, comparing them to matched controls on structural reasoning, standard test scores and overage rates.

3. Integration with DSM-H and decolonial frameworks. Examine whether children who grow up with Mandombe-based symbolic models show different profiles on DSM-H–aligned measures of structural antisociality, colonised-mind scales and math anxiety, compared to peers educated exclusively with imported scripts and symbols.

A fourth line would examine teacher experience more systematically: does Mandombe reduce cognitive load and verbal effort when explaining part–whole relationships, and does it change how teachers themselves think about mathematics?

---

6. Conclusion

This study does not claim that Mandombe magically produces mathematical prodigies. It does not attempt to generalise from a single small sample to the whole Congolese school system. What it does show is more modest and, in some ways, more fundamental: Mandombe’s branch-based geometry can function as a concrete early-childhood equation model.

When young children learn arithmetic through mvuala shapes and their branches, they quickly begin to talk about “complete” and “incomplete” shapes, “missing branches,” and “two mvuala with three branches each.” Structural explanations of part–whole and repeated-group relationships become common, rather than the rare achievement of a few outliers. When children learn the same content through standard dots and digits alone, such explanations remain fragile and concentrated in a minority.

In a context where most children are overaged and where math often feels abstract and punitive, this matters. It suggests that choosing and designing symbolic systems is not a neutral technical decision but part of the ethical and cognitive architecture of education. Mandombe, used carefully, does more than script words: it draws visible equations into the shapes children write, giving them a durable mental model they can carry into more advanced mathematics. The dramatic accelerations reported elsewhere may not happen everywhere or for everyone, but the mechanisms that could support them are already visible in the smallest hands tracing branches on a page.

---

[1] Nsiangani, K. (2010). Pan-Africanism Reimagined: Cognitive, Educational and Institutional Foundations. MEN-D Series on Decolonial Epistemology and Education, Vol. 1. Département Mandombe, Épistémologie & Neuro-Innovation, Université Simon Kimbangu / Programme Mandombe, Kinshasa (DRC).

[2] Nsiangani, K. (2014). The Dark Tetrad Traits of Empire: Structural Psychopathology and Colonial Governance. MEN-D Clinical & Political Psychology Monographs, Vol. 2. Département Mandombe, Épistémologie & Neuro-Innovation, Université Simon Kimbangu / Programme Mandombe, Kinshasa (DRC).

[3] Nsiangani, K. (2016). From Mvemba Nzinga to Modern Puppets: Cultural Violence and Colonial Leadership Pathologies in Congo. MEN-D Historical & Structural Pathology Series, Vol. 3. Département Mandombe, Épistémologie & Neuro-Innovation, Université Simon Kimbangu / Programme Mandombe, Kinshasa (DRC).

[4] Nsiangani, K. (2021). Early Childhood Math Learning Through Mandombe: Cohort Evidence and Curriculum Foundations. MEN-D Curriculum & Research Reports, Série Pédagogie Mandombe, Vol. 5. Département Mandombe, Épistémologie & Neuro-Innovation, Université Simon Kimbangu / Programme Mandombe, Kinshasa (DRC).

[5] Nsiangani, K. (2021). Curriculum MEN-D: Programme de Littératie et Mathématiques en Mandombe (Préscolaire–Primaire). Département Mandombe, Épistémologie & Neuro-Innovation, Université Simon Kimbangu / Programme Mandombe, Kinshasa (DRC).

[6] Nsiangani, K. (2023). Toward DSM-H: Diagnostic and Statistical Manual of Human Structural Pathologies. MEN-D Clinical Frameworks Series, Vol. 7. Département Mandombe, Épistémologie & Neuro-Innovation, Université Simon Kimbangu / Programme Mandombe, Kinshasa (DRC).

[7] Mbiti, J. S. (1990). African Religions & Philosophy (2nd ed.). London: Heinemann.

[8] Menkiti, I. A. (1984). Person and community in African traditional thought. In R. A. Wright (Ed.), African Philosophy: An Introduction (3rd ed., pp. 171–181). Lanham, MD: University Press of America