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Where do we come from? What is the traditional paper-based mathematical recording in early childhood education?
For many years, we’ve asked whether it makes sense for young children to engage in “math activities” through paper tasks. Most worksheets, whether from publishers or created by schools, tend to be very closed-ended-clearly stating what the child is expected to do-and allow only one correct answer. Ideally, all students are expected to perform the same way in these activities.
We’re referring to activities like these:
If we focus on the mathematical content of these activities, it becomes clear that there are other ways to approach it that are more appropriate for the children’s age, interests, and developmental needs.
In the first example, for instance, we could identify flat shapes using cards or logic blocks, look for triangles on objects around the classroom, or construct them using sticks and line segments.
In the second example, we could form groups based on a number—having as many children group together as the number indicates, creating collections with hands-on materials, or playing card games that match numerals with quantities.
So, do we really need these worksheets? What value do they bring?
If we look more closely at what these worksheets actually ask of students, we’ll see that in the vast majority of cases, they involve identifying, recognizing, or reproducing — but rarely thinking. As a result, there’s little to no real mathematics involved. Most worksheets are closed-ended tasks with a single correct answer.
What’s more, if we examine the activity children engage in to complete the worksheet — like distinguishing triangles from non-triangles or counting a given number of items — we’ll find that most children can finish it quickly. So how is the time justified? Often, it’s by spending a long time coloring without going outside the lines — which, of course, has nothing to do with math.
Another important question is whether children have developed the fine motor skills necessary to carry out what’s being asked of them. This is something we already raised in the article “Number formation in Early Years education,” which concludes that we must allow children to mature in all the foundational areas involved before expecting precision in any kind of written record.
So, what’s the advantage of offering any paper-based tasks at all?
That they can provide a moment of calm, where each child has space to act and reflect individually and personally.
What is the methodological structure of the Workshop activitie
If we analyze how the Workshops are carried out, we’ll see that most of them begin with some kind of playful resource: movement games, stories, songs, art, challenges, short narratives, and so on. In this way, we create learning situations in cross-disciplinary contexts — that is, activities that engage multiple areas and subjects.
That’s why the initial activities are necessarily very collaborative — sometimes with the whole class, sometimes in small groups, or often in pairs. But doesn’t it ever happen that…
- You’re left wondering whether a particular child really grasped the core idea?
- You notice that some children fade into the background during group work because they’re not as outgoing as others?
- You feel that most children also need quiet moments to process, piece together, and express their ideas?
That’s why we almost always end the session with a more individual challenge — whether it involves manipulating materials or leaving some kind of record on paper. Of course, we always avoid the drawbacks we identified earlier in traditional worksheet use.
What is the educational rationale behind this sequence for learning mathematics in early childhood education?
We, inspired by researchers from various fields — such as Baroody (1988), Bishop (1999), Puig Adam (1960), and Bueno (2019) — and aligned with current curriculum guidelines, design learning situations. That means creating cross-curricular contexts through games — like tossing objects, asking how many land inside and how many outside, etc. — or through imagined scenarios based on songs and stories, in which we need to count, measure, locate, design, and explain (Bishop, 1999).
But we don’t stop there. Following Baroody’s guidance, we look for patterns and regularities in what we’ve experienced. As educators, we help children focus on the mathematical relationships we’ve uncovered — going beyond the materials or the original context — and begin to explore that content on a more abstract level by making new connections.
Thus, in line with Puig Adam, we help children develop their capacity for abstraction by focusing on mathematical relationships and shifting between different forms of representation. Here is where we connect with neuroscience (Bueno, 2019): the brains of children at this age are primed to build connections between different brain regions and to form new neural networks. The more diverse these connections, the more robust and expansive the networks become; and having more connections means having more knowledge — and being able to use it more efficiently.
Each shift in representation — whether it’s a change in materials, new challenges around the same mathematical idea, a drawing on paper, or a verbal explanation of what was represented — broadens and strengthens each child’s mental network connected to the concept being explored. These mental networks become — and will continue to be — the foundation for all related future learning.
Example of a learning situation in early childhood education
We’ll conclude this article with a concrete example of a sequence of representational shifts in early childhood, illustrated through a learning situation.
First Workshop
1. We create a cross-curricular learning situation using a basket toss game, where children are required to count the number of items that land inside and outside the basket. The game is engaging and fosters a learning environment that goes beyond mathematics — it also supports psychomotor and social skills, as children must agree on the order of turns, decide where to place the throwing line, and so on.
2. While they’re playing, we ask if they’d like to keep track of each player’s score as a way to remember them. This marks the first representational shift.
For example, the three ducks inside the box and the four outside are now recorded using numeric symbols, which help us remember the exact quantities in each case.
3. At the end of the workshop, we ask the children if they’d like to explain what they did today — what the activity was about. Then we can ask ourselves: Do their representations reflect the intended learning goals?
- Identifying pairs of numbers that add up to 7
- Using 7 as a reference number
4. And here comes the second representational shift — likely the most meaningful one — because it’s chosen by the children themselves.
In this case, the student voluntarily represents the two quantities in a schematic (pictorial) way and complements them with the appropriate numeric symbols. This connection between pictorial and symbolic representation is key to developing an understanding of the concept of number.
5. We conclude the task with a brief oral description of what the children have represented and, if possible, we ask follow-up questions to assess the activity.
This verbalization of what has been drawn or recorded on the sheet represents the third shift in representation.
Second Workshop
6. In the following workshop, we start with the scores from the previous session and use pairs of number cards to show and discuss our results, while asking ourselves which and how many pairs of numbers add up to 7 — this marks the fourth shift in representation.
Here, we support the children’s process of abstraction by helping them connect ideas and expand their neural networks. The new challenge is no longer just about their own individual scores, but also about everyone else’s — and perhaps even about discovering all the possible pairs of numbers that add up to 7.
7. There are also moments where we collaboratively build a mathematical sentence that expresses the relationships we’re discovering, using more symbolic language. This represents the fifth shift in representation.
8. As we continue forming pairs that add up to 7, there are times when we need to use tangible or countable materials — and that’s perfectly normal. Each child develops at their own pace, and it’s important to provide the resources they need to connect with the task. As a result, both the materials and the types of representation shift back and forth depending on individual needs.
9. Once again, we ask the children if they’d like to explain what they did today. What was the workshop about? What did we discover or learn? This marks the sixth shift in representation.
10. And sometimes, things like this happen:
This student is not only able to find all the number pairs that add up to 7, but she also comes up with her own reasoning to prove that she has found them all — something that hadn’t come up during the in-person session.
It’s important to note that finding all the decompositions of 7 into two addends and providing an argument to justify that all possibilities have been found is not the main objective of the session. We know that not all 5-year-olds are ready to reach this point.
However, this student is by no means an exception. In every group I’ve observed, there are always at least one or two children who connect the dots and solve the most complex challenge.
Just as we have tools to support less mature learners, we must also provide opportunities for those who are ready to go further.
Now, if you’d like, take another look at what those early math worksheets ask of children, and compare that with what we’ve just explored. Each of us can draw our own conclusions.
As for us, we’ll continue supporting this line of reflection with two more pieces. We began with Paper-Based Recording in Early Childhood Education: Yes or No, and Why? Next, we’ll continue with Sheets in Early Childhood: What and Why?, where we’ll explore the different types of individual tasks that can be used to close a cross-curricular activity.
And in the third installment, The Blank Page in Early Childhood: How and Why?, we’ll help you understand the scope of the approach and offer tools for its interpretation and assessment.
See you soon,
Mequè
References
Baroody, A. (1988). El pensamiento matemático de los niños. Visor
Bishop A. (1999). Enculturación matemática. La educación matemática desde una perspectiva cultural. Paidós.
Bueno, D. (2019). Neurociència para educadores. Octaedro.
Puig Adam, P. (1960). La Matemática y su enseñanza actual. Ministerio de Educación.
