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What is algebra? Abstract Thinking Development

Abstract Thinking Development: Psychological Shifts from Concrete to Formal Operations in Secondary Math

As a tutor working with secondary students in Bukit Timah, I’ve watched hundreds of kids navigate that tricky leap from Primary to Secondary math.

One moment they’re comfortable with bar models and concrete problem sums from PSLE; the next, they’re staring at algebraic equations or coordinate geometry, wondering why nothing feels tangible anymore.

Parents often come to me worried—”My child was fine in Primary, but now they say math doesn’t make sense.” It’s a common story, and it ties directly to a big psychological shift happening in adolescence: the move from concrete to formal operational thinking.

Jean Piaget, the Swiss psychologist whose work still shapes how we understand cognitive development, described this beautifully.

Up to around age 11 or 12—right around the Primary to Secondary transition—most children operate in the concrete operational stage. They think logically, but mostly about real, hands-on things.

That’s why Singapore’s Primary math syllabus works so well with visuals: bar models for ratios, manipulatives for fractions, diagrams for everything. It matches how their brains are wired at that age.

Then, during secondary years (roughly 12–16), many students start entering the formal operational stage.

Here, thinking becomes abstract and hypothetical. They can reason about variables that represent unknowns, imagine “what if” scenarios, manipulate ideas in their head without needing physical props, and even ponder proofs or theoretical concepts.

In math terms, this is exactly what’s needed for Secondary E-Math and especially Additional Math: solving for x without a picture, understanding functions as relationships rather than just plots, or grasping rates of change in calculus.

But here’s the catch—not every student makes this shift at the same pace. Piaget and his collaborator Bärbel Inhelder noted that while the potential emerges in adolescence, it needs the right environment and experiences to fully develop.

Recent studies from 2023–2025 echo this. For instance, research on adolescent brain development shows the prefrontal cortex (responsible for planning and abstract reasoning) is still maturing well into the late teens.

A 2024 meta-analysis in educational psychology journals found that students who struggle with algebraic abstraction often haven’t fully transitioned yet, leading to frustration in early Secondary years.

In my small-group classes—usually just three students so no one can hide—I see this play out every week.

Take a Sec 1 student from a neighbourhood school who breezed through PSLE with models but freezes when we introduce linear equations. “Where’s the bar?” they ask.

Or a brighter one from an IP school who intuitively gets variables but struggles to explain why.

And then there are those in between, maybe dealing with the added pressure of Full Subject-Based Banding, trying to accelerate into G3 topics.

What helps bridge the gap? Patience, first of all. Enough contact time to build trust—kids open up when they feel safe asking “silly” questions.

I’ve had students from all walks: top schools pushing for early A-Math, others catching up after a shaky Primary foundation.

The ones who click fastest are those where we build chemistry. They start seeing me as a reliable guide, someone they can admit confusion to without judgment.

Practically, we ease the shift using what MOE already encourages: the Concrete-Pictorial-Abstract (CPA) progression. Even in Secondary, I revisit concrete examples before jumping to pure symbols.

For instance, when teaching quadratic graphs, we might start with real parabolas—like projectile motion from a basketball throw—then sketch, then manipulate equations.

Interleaved practice helps too: mixing topics so they learn to flexibly apply abstract rules. And always, explain-back: I make them teach the concept to their peers in the group. It forces formal thinking.

That said, not everyone needs the same push. Some kids arrive already thinking hypothetically—great for proofs in A-Math.

Others need more scaffolding. With consistent guidance, though, the shift happens. I’ve seen quiet Sec 2 students who started avoiding algebra turn into confident Sec 4s tackling differentiation smoothly.

It’s rewarding because once formal operations kick in, math stops feeling like rote rules and becomes a way to explore possibilities.

Concrete vs. Abstract Thinking in Math: Why Algebra Feels Abstract (When It Doesn’t Have To)

Parents often ask me, “Why does my child suddenly hate math in Secondary 1 when they loved it in Primary?”

The answer usually lies in how we present algebra—most students think it’s this big leap into “abstract” territory, full of meaningless letters and rules.

But here’s the truth I’ve seen over years of teaching kids from every kind of school: at its foundation, algebra is actually very concrete.

The real issue is that the bridge from Primary’s hands-on math isn’t built strongly enough, so when things get more advanced, the abstract side feels overwhelming.

Let me break it down with clear examples.

Concrete Thinking in Math (Primary Style)

Concrete thinking means working with things you can see, touch, or directly picture. Singapore’s Primary math excels here because of the CPA (Concrete-Pictorial-Abstract) approach.

Example: A classic PSLE ratio problem
“John has 12 marbles. He has 3 times as many marbles as Mary. How many marbles does Mary have?”

Concrete way most Primary kids solve it:
They draw bars. John’s bar is split into 3 units, total 12 marbles → 1 unit = 4 → Mary has 4 marbles.
Everything is visual, tangible. No letters needed. The child “sees” the relationship physically.

This is pure concrete operational thinking—logical, but tied to real representations.

Algebra at Foundation Level Is Actually Concrete Too

Now take the same problem in Secondary 1 algebra form:
“Let Mary have x marbles. John has 3x marbles. John has 12 marbles. Find x.”

To many Sec 1 students, this looks abstract—”Why use x? What’s the point?”
But if we make the connection properly, it’s just as concrete as the bar model.

Strong concrete bridge I use in class:

  • Start with the bar model exactly like Primary.
  • Then label the small unit as “x” right on the bar.
  • Show that 3x = 12, so x = 4.
  • Point out: “See? The ‘x’ is just a label for the length of Mary’s bar. It’s the same thing you drew before—only now we’re writing it efficiently.”

When I do this step-by-step with my small groups, the lightbulb moment happens fast.

A quiet Sec 1 boy from a proper G3 secondary school once said after this, “Oh… so x is actually the bar?” Exactly. At this level, algebra isn’t abstract—it’s a shorthand for the concrete picture they already understand.

Where It Starts Feeling Truly Abstract

The shift happens when we move to more advanced topics and the concrete links aren’t reinforced enough.

Example: Solving simultaneous equations or quadratic equations
“Two numbers add to 10 and multiply to 21. What are the numbers?”

Early Secondary often jumps straight to:
Let the numbers be x and y
x + y = 10
xy = 21
Then substitution or quadratic: x(10 – x) = 21 → x² – 10x + 21 = 0

Without strong prior connection, students see only symbols manipulating symbols—no meaning. It feels abstract and arbitrary.

But if we’ve built the foundation well:

  • They remember x and y as actual quantities (like lengths of bars).
  • They can sketch two bars adding to 10 units, area 21, and see the quadratic as describing that relationship.
  • Suddenly, even factoring (x – 7)(x – 3) has meaning—those are possible pairs.

In A-Math, when we hit calculus or abstract proofs, the gap widens further if early connections were weak. Rates of change feel detached if students never deeply linked variables to real quantities.

Why Move from Models to Algebra? The Real Reasons Behind the Shift

Parents sometimes ask me this directly: “Bar models worked so well for PSLE—my child scored AL1 using them. Why complicate things with algebra in Secondary? Why not just stick to models all the way?”

It’s a fair question. If concrete visuals get the job done, why introduce letters and equations that seem to confuse everyone at first?

Here’s the honest answer I’ve come to after teaching Secondary and A-Math to every kind of student for a long time: we move to algebra not because models stop working, but because certain problems become impractical—or even impossible—to solve efficiently with models alone. Algebra is the natural next tool when the questions get more complex, more general, or involve unknowns in multiple places.

Let me explain with examples that I often use in my small groups.

1. Problems Get Too Messy for Models

Take a straightforward Sec 2 word problem:
“Two years ago, the father was 5 times as old as his son. In 8 years’ time, the father will be 3 times as old as his son. How old is the son now?”

With bar models, you can still solve it—but look at the diagram: you’d need bars extending left (past) and right (future), multiple units, gaps for the time differences. It gets crowded fast. Many students draw it wrong the first few times, and checking is hard.

With algebra:
Let son’s current age be x.
Father now: 5(x + 2) – 10 wait, no—simpler:
Two years ago: father = 5 × son
So father now = 5(x – 2) + 2, but we usually set:
Son now: x
Father now: 5(x – 2) + 2 = 5x – 8
In 8 years: father = 3(son)
5x – 8 + 8 = 3(x + 8)
5x = 3x + 24
2x = 24 → x = 12

Clean, quick, and easy to check. Once the setup is correct, the mechanics are straightforward. The model still helps visualise the setup, but algebra handles the calculation without clutter.

2. Multiple Unknowns or Relationships

PSLE rarely has two independent unknowns. Secondary loves them.

Example:
“The length of a rectangle is 4 cm more than its breadth. Its area is 45 cm². Find the dimensions.”

Bar model possible, but already pushing it.
Algebra:
Breadth = x
Length = x + 4
x(x + 4) = 45 → quadratic.

Now imagine a Sec 3 simultaneous equations problem with speed, distance, meeting points—drawing bars for upstream/downstream or relative speeds becomes a nightmare. Algebra lets you write two equations cleanly and solve systematically.

3. Generalisation and Patterns

Primary math often deals with specific numbers. Secondary starts asking for general rules.

Example:
“Find the sum of the first n odd numbers.”
You can draw squares of dots and see it’s always n², but proving it for any n requires algebraic pattern spotting:
1 + 3 + 5 + … + (2n–1) = n²

Or number sequences in Sec 2—finding the nth term. Models help spot the pattern, but algebra expresses it precisely.

4. Preparation for Advanced Topics

Once you reach A-Math or even E-Math trigonometry and calculus, models alone won’t cut it. Functions, rates of change, differentiation—all rely on algebraic manipulation.

If students never get comfortable with symbols as flexible tools, calculus feels like magic instead of logic.

So Why Not Just Keep Models Longer?

We actually do—MOE’s CPA approach encourages keeping pictorials as long as useful.

In my classes, I often let P6 students sketch quick models first, if possible too, then translate to algebra. (algebra is now taking a bigger play in P5/P6 Math)

The goal isn’t to abandon concrete thinking; it’s to layer algebraic thinking on top so they have more tools.

Think of it like language:
Primary = speaking in full sentences with pictures to help.
Secondary = learning grammar and writing so you can express more complex ideas clearly and efficiently.

The students who struggle most are usually those who were rushed through the translation step. They either
(a) never saw how algebra connects to their trusted models, or
(b) were drilled on algebraic procedures without meaning.

That’s why in my small groups I insist on the bridge if possible during P6-S1 December holidays: solve with model → solve with algebra → compare both answers → explain why they match.

After a few weeks, most students start preferring algebra because it’s faster and less error-prone for harder questions.

Bottom line: we move to algebra because the problems grow up. Models are fantastic foundations, but algebra gives students the power to tackle questions that models alone would make exhausting or impossible.

When taught with strong concrete links, it doesn’t feel like a leap—it feels like the next logical step.

If your child is in that transition phase and finding algebra frustrating, a bit of patient bridging usually turns things around quickly. Happy to chat if you’re seeing this at home.

What I’ve Learned from Teaching All Kinds of Students

Kids from top schools often make the leap faster because they’ve seen more patterns early. But even they sometimes lack deep understanding if it was rushed.

Students from other schools might take longer, but with patient scaffolding—revisiting concrete models even in Sec 3 or 4—they catch up beautifully and often surpass.

The key is time and trust. When a student feels comfortable saying “I don’t get why we use letters,” we can go back to bars, objects, or real-life contexts (speeds, distances, money) until it clicks.

Once that concrete-to-symbol link is solid, advanced algebra stops being scary and starts making sense.

Algebra isn’t inherently abstract at the start—it’s an extension of the concrete thinking Primary math builds so well.

The problem comes when we don’t explicitly show the connection, letting the gap grow until advanced topics feel like a different world.

If your child is struggling with this shift—whether in Sec 1 basics or pushing into A-Math—a small-group setting where we can revisit foundations without judgment often makes all the difference. Feel free to drop me a message if you’d like to talk about it.

If your child is hitting this wall—maybe frustrated with abstracts in Sec 1 algebra or gearing up for more in A-Math—a structured small-group setting can make all the difference.

We focus on building that bridge patiently, turning potential stumbling blocks into strengths. Feel free to reach out if you’d like to chat about how this fits your situation.

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5 internal links (relevant “next reads”)

5 research papers closely related to abstract thinking + the Sec Math transition

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