When we watch a child trying to learn something they cannot yet do, we often focus on what is missing.
They cannot recognise letters. They cannot solve a calculation. They cannot answer a question. They cannot complete an activity independently.
These observations can help us identify difficulties, but they can also lead us to a premature conclusion: that the child is not learning.
And when results do not appear, something else can happen: we, as adults, may begin to lose patience.
We explain again, repeat instructions, increase the number of exercises or look for other ways to get the child to respond.
Sometimes we feel frustrated because, despite the time and effort we have invested, we cannot see the progress we expected.
Throughout my work as a psychologist and educator, I began to question this way of interpreting learning.
What happens before a child can demonstrate a new capability for the first time? And how do we, as adults, respond when we cannot yet see that progress?
These questions led me to pay particular attention to what I describe as n=0.
1. What does n=0 mean?
In my understanding of learning, I use n to represent a capability that a child can already perform independently, without an adult guiding every step or providing the answer.
When we refer to n=0, we are describing a situation in which we cannot yet observe the first independent response or action associated with the specific capability we want to develop.
I call this period “silent learning” because the child may be observing, listening, attempting or beginning to understand, even though they cannot yet demonstrate what they have learned independently.
For example, imagine a child who is beginning to learn mental maths and cannot answer any calculations correctly in an assessment completed without assistance.
In that activity, their observable result is zero.
But this does not mean the child knows nothing about mathematics, has no other capabilities or is unable to learn.
It simply means that the child has not yet demonstrated that specific capability independently, under the conditions of the activity.
They may understand part of an explanation or complete some steps with guidance, while still being unable to produce an independent response. Assisted participation can be part of learning, but it is not the same as having already acquired the independent capability represented by n.
A result of zero tells us what we were able to observe at a particular moment. It does not tell us everything the child knows, understands or may be beginning to learn.
2. Learning can begin before we can observe it
Think of a child learning a new word.
They hear it for the first time, look at a picture associated with it and encounter it again during different activities.
Throughout this process, they may still be unable to recognise or say the word independently when asked.
Does that mean no learning has taken place?
Not necessarily.
The child may be becoming familiar with the sound, beginning to notice its characteristics or connecting it with previous experiences.
However, we also need to be careful: we cannot assume that a capability is developing simply because a child is listening or observing.
We need to provide opportunities for the child to participate and gradually demonstrate what they have learned.
This is why I believe it is important to distinguish between two moments:
Learning that may be taking place, even though we cannot yet demonstrate it.
And:
The first observable independent response that allows us to recognise progress.
This distinction helps us avoid two mistakes.
The first is concluding too quickly that the child is not learning.
The second is assuming that a capability is developing without having information that allows us to recognise progress.
3. Adults’ impatience when we cannot see results
There is something I consider particularly important: how difficult it can be to wait when we are teaching and cannot yet see progress.
Parents and teachers invest time and effort in children’s learning. We explain, demonstrate, repeat instructions and look for different ways to help.
Naturally, we expect these experiences to produce results.
But what happens when the child continues to struggle to answer correctly on their own?
We may begin to feel impatient, worried or frustrated. Perhaps we think the child is not paying attention, is not trying hard enough or does not want to learn.
We may even begin to question our own ability to teach them.
This can lead us to intervene more and more. We repeat explanations, increase the number of exercises, correct every mistake immediately or end up completing part of the activity for the child.
Without realising it, our need to obtain an answer can begin to shape the learning experience.
And children can sense our impatience. They may notice our frustration when they make another mistake or feel the pressure of being expected to provide an answer they do not yet know.
In some cases, this can increase their insecurity or make them less willing to try again.
I remember how important it was for me to hear a child say:
“I don’t want to feel that I’m not capable.”
Those words helped me understand that when a child avoids an activity, they are not necessarily rejecting learning. Sometimes they are struggling with the experience of being unable to do something adults expect them to do.
This also requires us to observe our own responses.
Are we allowing time for a capability to begin developing, or are we demanding results before the child is ready to demonstrate them?
Patience does not mean waiting indefinitely without making changes. Nor does it mean assuming that learning is taking place when we have no signs of progress.
It means observing, reviewing the conditions, adjusting our guidance and providing suitable opportunities for that first independent response to emerge.
As adults, we also need to learn to tolerate the uncertainty that exists between teaching something and being able to observe that the child has learned it.
Not seeing immediate results can be frustrating. But our frustration should not become another difficulty for the child.
4. From n=0 to n=1: the importance of the first step
When a child moves from being unable to demonstrate a capability independently to producing a response they could not previously achieve on their own, something important has happened.
We can represent this first observable step as the transition from n=0 to n=1.
Imagine a child who cannot solve any of the maths calculations presented to them without help.
After participating in appropriate learning experiences, they manage to solve one calculation correctly and independently.
To an adult, that single answer may seem like a small achievement. The child may still be far from the knowledge and skills expected at their year level.
However, from their own starting point, a meaningful change has occurred.
Previously, the child could not demonstrate that independent response. Now they have achieved it for the first time.
It is important to distinguish the first achievement, n=1, from a consolidated capability. One correct answer provides initial observable evidence, but the child still needs opportunities to demonstrate that they can do it again independently and with sufficient consistency.
Only then can that capability become a reliable new n, a starting point for further learning.
In my experience, this first step deserves particular attention. It can also change how we support the child.
Instead of focusing exclusively on everything they still cannot do, we can recognise what they are beginning to achieve.
And that recognition can matter both to the child and to the adults supporting their learning.
5. What I learned from observing Bamboo results
Between 2016 and 2019, we analysed mental maths assessments from 10,872 students who participated in the Bamboo programs.
The assessments lasted two minutes, were completed without adult assistance and allowed us to observe how many calculations each child answered correctly.
The results led me to reflect on how we interpret progress.
Consider an example: a child who initially answers no calculations correctly and, in a later assessment, manages to solve five independently.
Under the classification criteria, the child may still remain in the Low category.
If we look only at that classification, we might conclude that nothing has changed.
However, when we examine the number of correct answers, we find an important difference.
The child has progressed from solving no calculations correctly to solving five without help.
This progress does not mean the child has reached all the expected learning goals, nor can we attribute the result solely to the program.
It also does not tell us everything about the child’s mathematical capabilities.
But it shows something worth recognising: under those assessment conditions, the child was able to demonstrate independent responses they had not previously demonstrated.
This experience strengthened my interest in observing learning from each student’s individual starting point, rather than relying exclusively on an overall classification.
It also made me think about the importance of recognising progress for the adults who support children’s learning.
When an adult sees only that a child remains in the Low category, they may feel that all their effort has been unsuccessful.
But when they understand that the child has moved from zero to five correct answers, they have different information to consider.
The final result may still be far from what is expected, but that does not mean the child has made no progress.
Recognising this difference can help us make better decisions and support learning with more realistic expectations.
6. What can we change when the first response has not yet appeared?
When a child cannot complete an activity, we may think they need to try harder, pay more attention or repeat the exercise.
But before reaching that conclusion, I believe it is important to examine the learning conditions.
We can ask ourselves:
- Instructions: Does the child understand exactly what they are expected to do?
- Starting point: Does the activity require previous capabilities the child has not yet developed?
- Difficulty: Are we presenting a challenge that is too great?
- Time: Does the child have enough time to observe, think and respond?
- Materials: Could a demonstration or visual support help the child understand?
- Practice: Has the child had suitable opportunities to try and practise again?
- Response: What changes when we adjust any of these conditions?
I would also add a question directed at ourselves:
Are we adjusting the activity because we have observed what the child needs, or because we are becoming impatient when we do not see results?
This distinction matters.
Sometimes we insist on an activity because we believe that repeating it more often will resolve the difficulty. At other times, we constantly change strategies because we find it difficult to wait.
But neither endless repetition nor changing activities at every moment guarantees learning.
We need to observe what is happening and decide what may help the child take the next step.
Sometimes a small adjustment allows the child to begin participating. In other situations, we may need to reconsider the starting point, introduce simpler activities or provide specialised support.
We must also recognise that the absence of a response may be related to communication difficulties, emotional factors, accessibility needs or other circumstances.
For this reason, n=0 is not a diagnosis or an explanation of why a child is not yet responding.
It is a way of recognising that we have not yet observed a particular independent capability and that we need to understand what might help the child develop it.
7. Recognising the first step also matters in the classroom
In a classroom, children can have very different starting points.
While some complete an activity with ease, others may only be beginning to develop the capabilities needed to participate.
Teachers need to organise learning for the whole group, but they also need to recognise individual progress.
Imagine a mental maths activity.
A child who can already solve several calculations may be ready for new challenges. Another child may be learning to solve their first calculation correctly.
Both can participate in a shared learning experience, even though their individual progress is different.
The common learning goal helps organise the group’s work. Observing each child’s individual starting point helps us recognise what they are developing.
This does not mean that all children should receive exactly the same activities or that a shared learning experience will meet every child’s needs.
It means that we can value individual progress while preserving opportunities to learn together.
It also reminds us that progress that appears small when compared with the group may be very important to the child who has just achieved it.
8. The first achievement is not the end of learning
Recognising the transition from n=0 to n=1 does not mean stopping after the first correct response.
On the contrary, that achievement gives us an opportunity to develop and verify a new capability.
The child needs opportunities to repeat the activity, remember what they have learned, recognise mistakes and practise at different times.
Over time, that first response may develop into a skill the child can use independently, with greater accuracy, confidence and consistency.
From that point, we can introduce a new challenge.
This is the relationship between n=0, n=1 as the first observable achievement, and n+1 as the next challenge.
The first directs our attention to what happens before the first independent response. The next recognises that first step, while practice helps consolidate the capability before it becomes a reliable starting point for further challenges.
Both concepts remind us that learning is not simply about completing activities or achieving a particular grade.
Learning involves developing capabilities we previously could not demonstrate or use in the same way.
Conclusion: learning to recognise what is beginning to develop
For many years, education has relied on assessments to identify what students know and what they still need to learn.
These assessments are important.
But we also need to ask what happens when a child cannot yet demonstrate the capability we expect to observe.
Are we recognising their starting point?
Are we providing suitable experiences to help them develop that first capability?
Can we identify their progress, even when they have not yet reached the expected result?
And also:
How are we, as adults, responding when learning does not progress as quickly as we expected?
Impatience and frustration can arise when we teach and cannot see results. These reactions are understandable, particularly when we feel responsible for helping a child who is experiencing difficulties.
But we need to prevent our urgency to obtain an answer from stopping us from observing what is actually happening.
The concept of n=0 invites me to look at learning from a different perspective.
It does not mean ignoring difficulties or assuming that all children will learn in the same way.
It means recognising that the absence of an observable independent response does not define all of a child’s capabilities or determine what they may be able to develop.
It also invites us to value the first step, however small it may seem.
Because before solving five calculations, a child may have needed to learn how to solve one.
And before that first correct calculation, there may have been a process of observation, understanding and attempts that we did not yet know how to recognise.
We cannot always see the moment learning begins. That is why, alongside providing appropriate learning opportunities, we as adults need to learn to observe, adjust our expectations and recognise the value of the first step forward.