Manipulatives for Struggling Learners: What Works

Why Are Manipulatives Important In Math For Struggling Learners?

Yes, when they are used to teach a specific idea and linked to the written math; no, when they are used as a fun extra in place of teaching. Both findings come from careful research, and they fit together.

• The case for: the 2021 IES practice guide for elementary math intervention gives its strongest evidence rating to using concrete and semi-concrete representations, and tells teachers to connect them explicitly to mathematical notation.

• The caution: a study of first graders by Paul Morgan, George Farkas and Steve Maczuga, published in Educational Evaluation and Policy Analysis and reported by Education Week in June 2014, found that children with math difficulties gained most from explicit, teacher-directed instruction with practice. Manipulatives, calculators, music and movement used as general classroom activities did not help them.

• How the two fit: manipulatives work as part of explicit teaching, held by an adult who names what the blocks show, not as a replacement for it.

The need is real. In the 2024 NAEP mathematics results, 76% of US fourth graders performed at or above NAEP Basic, which means roughly one in four did not. Many of those children are missing a concrete picture of what the numbers mean, which is exactly the gap manipulatives can fill.

One more correction worth making: manipulatives do not work because some children are "hands-on learners". The research does not support matching teaching to a learning style. They work because quantity, place value and fractions are easier to understand when every child can see and move them.

What Are Concrete, Semi-Concrete, And Abstract Representations?

Concrete, semi-concrete and abstract are the three stages of the CRA sequence, the most widely used way to teach with manipulatives. The IES guide defines them this way:

• Concrete: three-dimensional objects or actions the child can move, such as regrouping with base-ten blocks or comparing two fractions with fraction tiles.

• Semi-concrete (representational): two-dimensional pictures, such as tally marks, dot drawings, strip diagrams, arrays and number lines.

• Abstract: numbers and symbols only, such as $4 \times 4 = 16$.

The stages are not three separate lessons. The guide's advice is to show the concrete model, the drawing and the equation together, so the child sees that all three describe the same thing. Singapore-style programs call this the concrete-pictorial-abstract approach, and Singapore math is built around it.

Which Manipulatives Work Best For Each Skill?

Pick the manipulative that matches the skill your child is stuck on, and use that one consistently. The table draws on the IES guide's own examples.

Counters And Ten Frames

Counters are anything small and identical: beans, buttons, plastic discs. They suit the earliest skills, from one-to-one counting to adding and subtracting within 20. Placed in a ten frame (a two-by-five grid), they show at a glance how many more make 10.

Base-Ten Blocks

Base-ten blocks come as units (1), rods (10), flats (100) and large cubes (1,000), and every size is exactly ten of the one below. That proportion is the point: the IES guide stresses that the ones should be one-tenth the size of the tens so regrouping is visible, not just a rule. They are the main tool for place value, regrouping in addition and subtraction, and later decimals.

Number Lines

A number line turns numbers into distances from zero, which makes comparing, adding on and fractions visible. The IES guide gives number lines a recommendation of their own. For a child who does not yet "get" the line, build it first from equal-length pieces, such as fraction tiles or rods laid end to end.

Fraction Tiles And Fraction Strips

Fraction tiles are bars cut into halves, thirds, quarters and so on, all from the same whole. Laying them side by side answers questions that rules cannot, like why $\frac{1}{3}$ is bigger than $\frac{1}{4}$. Use them before shortcuts such as cross-multiplying; the IES panel's view is that cross-multiplying is fast but does not build understanding of fraction size. If the words themselves are shaky, the pages on the numerator and the bottom number of a fraction are a quick refresher before the tiles come out.

Cuisenaire Rods

Cuisenaire rods are ten coloured rods with lengths from 1 to 10 units, each length its own colour. Because any rod can be named "the whole", the same set can show $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{6}$ of one length, and line up with a number line to show equivalent fractions.

Two-Colour Counters

Two-colour counters are discs with a different colour on each side, often one colour for positive and one for negative. A positive and a negative counter together make a "zero pair", which turns $-5 + 3$ into a picture rather than a sign rule. They suit older children who are stuck on negative numbers.

How Do You Use Manipulatives In Math So They Actually Teach?

A manipulative teaches when the child does the moving, the adult connects each move to the written math, and the child explains what happened. Without the connection it is just a toy.

1. Choose one tool for one skill. Base-ten blocks for regrouping, fraction tiles for comparing fractions. Do not bring out three tools at once.

2. Write as you build. Every time a block moves, write the matching number or step beside it.

3. Let your child do the moving. You ask, they build. "Show me 46 with blocks" teaches more than watching you do it.

4. Ask for the explanation. "Why did you trade those ten units?" is the question that turns building into understanding.

5. Draw the same problem. Before the blocks go back in the box, sketch what they showed.

The IES guide also warns that a few uses are not enough. Children need many chances to use the same tool before it becomes a "thinking tool" they reach for on their own.

Examples Of Manipulatives For Struggling Learners

These six examples show the concrete step, the written step that goes beside it and the answer, from early addition to fraction multiplication.

Example 1

Add $46 + 27$ with base-ten blocks.

Build 46 as 4 rods and 6 units, and 27 as 2 rods and 7 units. Combine the units:

$$6 + 7 = 13$$

Thirteen units is more than ten, so trade 10 units for 1 rod. That leaves 3 units and adds a rod to the tens:

$$4 + 2 + 1 = 7 \text{ tens}$$

$$46 + 27 = 73$$

Example 2

Which is bigger, $\frac{1}{3}$ or $\frac{1}{4}$?

The tempting answer: 4 is bigger than 3, so $\frac{1}{4}$ must be bigger than $\frac{1}{3}$.

Lay the tiles under a whole bar and the answer falls apart. Three thirds fill the bar, so each third is a long piece; four quarters fill the same bar, so each quarter is shorter.

$$\frac{1}{3} > \frac{1}{4}$$

The bottom number tells you how many pieces the whole is cut into. More pieces means smaller pieces, and the tiles make that visible before any rule is needed.

Example 3

Subtract $52 - 18$ with base-ten blocks.

Build 52 as 5 rods and 2 units. You cannot take 8 units from 2, so trade 1 rod for 10 units: now there are 4 rods and 12 units.

$$12 - 8 = 4$$

$$4 - 1 = 3 \text{ tens}$$

$$52 - 18 = 34$$

Check by adding back: $34 + 18 = 52$.

Example 4

Show $-5 + 3$ with two-colour counters.

Lay out 5 negative counters and 3 positive counters. Each positive-negative pair makes zero, so remove 3 zero pairs.

$$-5 + 3 = (-2) + (-3 + 3) = -2 + 0 = -2$$

Two negative counters are left, so the answer is $-2$.

Example 5

Use Cuisenaire rods to add $\frac{1}{3} + \frac{1}{6}$.

Call the 6-unit rod "the whole". Then the 1-unit rod is $\frac{1}{6}$, the 2-unit rod is $\frac{1}{3}$ and the 3-unit rod is $\frac{1}{2}$. Put the 2-unit and 1-unit rods end to end; together they match the 3-unit rod.

$$\frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$$

The rods show why $\frac{1}{3}$ has to be rewritten as $\frac{2}{6}$ first: two 1-unit rods are exactly the length of one 2-unit rod.

Example 6

Find $\frac{1}{2} \times \frac{2}{3}$ by folding paper.

Fold a square of paper into thirds one way and shade 2 of the 3 strips. That is $\frac{2}{3}$. Now fold it in half the other way and shade half of the shaded part twice over. The paper is now divided into 6 equal rectangles, and 2 of them are double-shaded.

$$\frac{1}{2} \times \frac{2}{3} = \frac{1 \times 2}{2 \times 3} = \frac{2}{6} = \frac{1}{3}$$

This is the area model the IES guide recommends for multiplying two fractions less than 1, where a number line is hard to read.

When And How Should You Fade Manipulatives?

Fade manipulatives once your child is accurate and can explain the idea with them, and not before. The IES panel's advice is direct: only fade concrete and semi-concrete representations as students become accurate doing the work abstractly, and revisit them whenever understanding wobbles.

Signs your child is ready to move on:

• They get the right answer with the tool most of the time.

• They can explain each move ("I traded ten units for a rod because...").

• They start predicting the answer before building it.

• They can draw the model quickly instead of building it.

A simple fading ladder:

1. Build and write together. Blocks and numbers side by side.

2. Draw instead of build. Quick sketches of rods and units, or a strip diagram.

3. Numbers first, drawing to check. Your child solves in numbers, then sketches to confirm.

4. Numbers only. The tools stay on the shelf, within reach.

5. Revisit on purpose. Bring the blocks back for one session when a new topic, such as decimals, builds on the old one.

Fading is not a one-way door. A child who could do two-digit subtraction in June may need the blocks again for three-digit subtraction with zeros in September, and that is the method working, not a step backwards.

Should You Make Your Own Manipulatives Or Buy Them?

You can make most manipulatives at home for almost nothing, and homemade versions work as long as they keep equal sizes. The one place it is worth buying is base-ten blocks or fraction tiles for a child who needs exact proportions to see the idea.

The IES guide raises a fair caution about home-made materials: finding or making good materials is harder than it looks, so follow a clear sequence of skills rather than collecting tools. A homemade set used in the right order beats an expensive kit used at random.

What Signs Show Your Child Needs A Concrete Step?

A child needs to go back to objects when they can follow a procedure but cannot say what the numbers mean. These are the signs to watch for:

• They count every addition from one, even for small numbers.

• They "borrow" in subtraction but cannot say what the borrowed 1 is worth.

• They think $\frac{1}{8}$ is bigger than $\frac{1}{6}$ because 8 is bigger than 6.

• They line up decimals by the last digit, as if 0.5 and 0.25 were whole numbers.

• They apply a rule perfectly on Monday and cannot start the same problem on Friday.

The last one is the classic sign of the memorizer, the child who learned the steps without the picture behind them. Many of these trace back to the same few early gaps, which is why fractions feel so hard for so many children who did fine with whole numbers.

Where Do Manipulatives Go Wrong For Struggling Learners?

Most problems with manipulatives come from how they are used, not from the tools themselves.

Mistake 1: Letting The Blocks Become Toys

Where it slips in: A new set of blocks arrives and building towers takes over.

Don't do this: Hand over the tool and hope the math appears.

The correct way: Give a few minutes of free play, then a clear job: "Show me 34." The IES guide names this exact problem, children who "just play" with concrete representations, and its fix is to teach how the tool is meant to be used.

Mistake 2: Never Writing The Numbers Down

Where it slips in: The child solves happily with blocks, and the session ends there.

Don't do this: Leave the concrete and the written math in separate worlds.

The correct way: Write the matching step beside every move. The first instinct when a child gets an answer with blocks is to celebrate and stop; the step that makes the learning transfer is writing $6 + 7 = 13$ next to the 13 units before the trade.

Mistake 3: Switching Tools Too Often

Where it slips in: Counters on Monday, a number line on Tuesday, a new app on Wednesday.

Don't do this: Change the model every time the child struggles.

The correct way: Stay with one model per skill, and match what your child's school or curriculum uses where you can. The IES panel calls consistency across classes and grades "critical" for struggling students.

Mistake 4: Fading Too Early, Or Never Fading

Where it slips in: Pressure to "catch up" pulls the blocks away; or comfort keeps them out for a year.

Don't do this: Remove the tool on a schedule, or let it become a crutch for answers your child can already reach.

The correct way: Fade when the readiness signs above are there, and bring the tool back when a new topic builds on the old one. The second-guesser who keeps rebuilding problems they can already solve needs a gentle nudge to the drawing stage, not more blocks.

When Should You Get Outside Help?

Look for outside help when manipulatives, used consistently and connected to written math for several weeks, still do not move a skill. Other signs:

• Your child cannot reliably tell which of two small numbers is bigger, or recognise small groups without counting, well after age 6 or 7.

• Basic facts do not stick despite steady, low-pressure practice.

• Math homework regularly ends in distress, while other subjects are fine.

These can be signs of a learning difference such as dyscalculia, or of anxiety, or of a gap a few grades back. Only a qualified professional, such as an educational psychologist or your child's school evaluation team, can assess for a learning disability, so start with your pediatrician or the school. Meanwhile, the guide on how to help a child with dyscalculia explains what support usually looks like, and a calm productive struggle is very different from the distress that signals a deeper problem.

How Does Bhanzu Help Children Who Struggle With Math?

Bhanzu begins with the idea behind a procedure before the procedure itself, which is the same order the CRA sequence follows: understand what the numbers mean, then work with the symbols. Every student starts at their real level rather than their grade label, so a child stuck on fractions because of a place-value gap gets the place-value work first.

Classes are live, online and small-group, for students from UKG through Grade 10, with peers from 20+ countries. Trainers ask children to explain their reasoning, which is the step that makes manipulative work at home pay off.

It fits families who are building concrete understanding at home and want a teacher to check the reasoning and fill earlier gaps. It does not fit a child who needs a diagnostic evaluation or specialist intervention for a learning disability; those come first, from a qualified professional.

Where Can Your Child Get Extra Help With Foundational Math?

If the concrete work at home has shown you a gap that needs a teacher, these pages match the most common situations:

• An elementary math tutor for counting, place value, regrouping and fractions.

• Math programs for elementary students who need a structured path through the foundations.

• An elementary math curriculum to see the order these skills are usually taught in.

• A middle school math tutor for fraction operations, negative numbers and ratios.

The Short Version

• Manipulatives for struggling learners work when one well-chosen tool is linked to the written math every time it is used.

• The CRA sequence moves from objects to drawings to numbers, and shows all three together while a child is learning.

• Base-ten blocks, fraction tiles, counters, number lines, Cuisenaire rods and two-colour counters each fit a specific skill.

• Fade a tool once your child is accurate and can explain it, and bring it back when a new topic builds on the old one.

• Homemade versions work well as long as the pieces are equal and the skills follow a clear order.

What Should You Try First?

Pick the one skill your child is most stuck on, choose the matching tool from the table, and run three short sessions this week using the five steps above: build, write, explain, draw. Keep a note of which moves your child can explain on their own. If you want a teacher to check your child's reasoning and find the earlier gap, Book a free demo class with a Bhanzu trainer.

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