Teaching Mental Maths Strategies in a Montessori Classroom: Ages 6–12

I know many Montessorians who love the Montessori maths materials. I do too.

One of the strengths of the Montessori maths materials is the way they make abstract concepts real. Take the fraction circles, for example. They allow a child to touch and experiment with different amounts, discover equivalent fractions and physically see how one fraction differs from another.

However, one area I feel we need to think about more carefully in our Montessori maths teaching is how we teach children to use mathematical strategies.

Montessori materials can help children understand what is happening mathematically, but using a material successfully does not always mean that a child can select an efficient strategy or transfer what they know into a different context.

What is a maths strategy?

Just like maths knowledge, a strategy needs to be taught. Children also need opportunities to solve problems using different strategies and to talk about how they reached an answer.

A strategy is something a child does to work out an answer. Once the child simply knows the answer without needing to work it out, it has become number knowledge rather than a strategy.

For example, a younger child may count four yellow beads one by one to find out how many there are. Eventually, the child will recognise the yellow bead bar as four without counting it. At that point, recognising four is no longer a strategy. It is knowledge.

As in other areas of Montessori learning, a younger child or a child at an earlier stage may be more reliant on equipment. An older child, or a child using a more advanced strategy, may be able to work the problem out mentally.

Children need to transfer their knowledge

Let’s take a child who knows that 6 × 4 is 24.

Imagine that this child is laying out six groups of four yellow beads on the checkerboard. Instead of using the multiplication fact they already know, the child counts each group:

4, 8, 12, 16, 20, 24.

This child may know the multiplication fact in one context, perhaps when using the multiplication charts, but may not yet transfer that knowledge to a different material or situation.

As teachers, we can help children make these connections. We might pause and ask:

• Do you already know six groups of four?
• Is there a multiplication fact that could help you?
• Can you see the same pattern on the checkerboard?
• Is there a quicker way to find the answer?

The aim is not to stop children using materials before they are ready. It is to help them recognise when their existing knowledge can make their work more efficient.

Connecting materials with mental strategies

A child might be given the equation 26 + 14 to solve with the Stamp Game.

After the child has solved it successfully with the material, we could show another way of thinking about the same equation.

The child might use place-value partitioning:

(20 + 10) + (6 + 4) = 30 + 10 = 40

They might also notice that 26 needs four more to reach the tidy number 30. The 14 can be split into 4 and 10:

26 + 4 = 30, then 30 + 10 = 40.

Both approaches connect directly with what the child has seen using the Stamp Game, but they also help the child begin moving towards mental calculation.

We do not need every child to use every strategy. However, we need to understand a range of strategies so we can recognise what a child is doing, help them see connections and decide what lesson might come next.

Children also develop their understanding of addition, subtraction, multiplication and division alongside each other and at different rates. A child may have a more advanced addition strategy than multiplication strategy, and this is normal.

The following is a quick, rather than exhaustive, look at some strategies children may move through. The lower down each list you go, the more complex the strategies generally become.

Addition strategies

• Counting objects from one: The child counts from one to work out how many objects there are. For example, they count each bead on a coloured bead bar.
• Grouping objects: The child puts objects into groups to make the quantity easier to count.
• Counting all: The child may use fingers, materials or mental counting to solve an equation such as 7 + 4. They count 1, 2, 3, 4, 5, 6, 7 and then 8, 9, 10, 11.
• Counting on: Instead of counting both amounts from one, the child begins with the larger number and counts on. For 7 + 4, they might say 7, 8, 9, 10, 11.
• Counting on with larger numbers: The child may solve 37 + 9 by counting 38, 39 and so on until reaching 46. They may still use fingers to keep track.
• Using doubles: The child uses a known double to solve a nearby fact. For example, 7 + 8 can be thought of as 7 + 7 + 1, or 8 + 8 − 1.
• Making a ten or tidy number: A tidy number is one that ends in zero. The child might solve 38 + 6 as 38 + 2 = 40, followed by 40 + 4 = 44.
• Place-value partitioning with two-digit numbers: The child might solve 32 + 24 as (30 + 20) + (2 + 4).
• Place-value partitioning with three-digit numbers: The child applies the same idea to hundreds, tens and units.
• Using tidy numbers and compensation: The child changes the numbers into easier amounts and compensates for the change. For example, 89 + 21 can be thought of as 90 + 20.
• Adding fractions: The child splits fractions and uses equivalent fractions. For example, 3/4 + 6/8 can be recognised as 6/8 + 6/8, which equals 12/8 or 1 1/2.
• Applying whole-number strategies to decimals: The child uses place value, partitioning, reversing or tidy-number strategies with decimal numbers.
• Converting between forms: The child converts between fractions, decimals and percentages before adding or comparing amounts.

You may notice that place-value partitioning can look different from the standard sequence used with some Montessori equipment, where children begin with the units and then move to tens and hundreds.

Children should not be discouraged from using place-value partitioning simply because it looks different from the material-based procedure. Instead, we can help them understand how both approaches represent the same mathematical relationship.

Subtraction strategies

• Removing objects from a group: The child physically takes objects away to find out how many are left.
• Counting back: The child may use fingers, materials or mental counting. For 12 − 3, they might count 11, 10, 9.
• Counting back with larger numbers: The child solves 56 − 8 by counting backwards from 56 until reaching 48.
• Using place value: The child might solve 734 − 106 as 734 − 100 − 6.
• Using tidy numbers and compensation: The child might solve 834 − 479 as 834 − 500 + 21.
• Reversing the equation: The child uses addition to solve subtraction. For example, 723 − 342 can be thought of as 342 + ___ = 723.
• Applying strategies to decimals: The child uses place value, reversing and tidy-number strategies with decimal amounts.

Some children find subtraction easier when it is presented as finding the difference rather than always taking away. Asking, “How far is it from 342 to 723?” can encourage the child to use addition rather than counting backwards over a large distance.

Multiplication strategies

• Repeated addition: This is often evident when a child uses the multiplication bead board. The child might work out 4 × 4 by calculating 4 + 4 + 4 + 4.
• Skip counting: The child counts in equal jumps, such as 4, 8, 12, 16.
• Using known multiplication facts: The child recalls a multiplication fact rather than rebuilding it each time.
• Reversing a multiplication fact: The child knows that 6 × 5 has the same answer as 5 × 6.
• Using doubles: The child may solve 6 × 8 by doubling 3 × 8, or solve 4 × 7 by doubling 2 × 7.
• Using place value and partitioning: The child solves 72 × 5 as (70 × 5) + (2 × 5).
• Using the distributive property: The child may solve 8 × 7 as (5 × 7) + (3 × 7).
• Using tidy numbers and compensation: The child might solve 19 × 6 as 20 × 6 − 6.
• Applying strategies to decimals: The child uses place value, reversing and compensation with decimal multiplication. This often develops later than the equivalent addition and subtraction strategies.

The multiplication bead board, coloured bead bars, multiplication charts and checkerboard all give us opportunities to help children connect repeated addition, skip counting, known facts and place-value strategies.

Division strategies

• Sharing or grouping with materials: The child physically shares a quantity into equal groups or works out how many groups can be made.
• Connecting division and multiplication: The child knows that a division fact is related to a multiplication fact. For example, 24 ÷ 6 can be solved by asking, “Six times what equals 24?”
• Repeated halving: The child might solve 48 ÷ 4 by dividing 48 by 2 and then dividing the answer by 2 again.
• Using place value: The child might solve 168 ÷ 7 by partitioning 168 into 140 and 28. Then 140 ÷ 7 = 20 and 28 ÷ 7 = 4, so 20 + 4 = 24.
• Using a nearby tidy number: The child may use a nearby compatible number and then compensate. For example, 208 ÷ 7 could be approached by considering 210 ÷ 7 and then thinking carefully about the effect of the extra two.
• Applying strategies to decimals: The child uses place value, multiplication knowledge and compensation to solve division problems involving decimals.

When using compensation in division, we need to be careful. Addition and subtraction compensation can often be adjusted quickly, but division does not always allow us to simply “take off the extra” without considering how that amount is being divided.

How to help a child choose a strategy

One of the most useful things we can do is ask the child how they worked something out.

Questions might include:

• Can you show me what you did?
• What did you notice?
• Did you use a fact you already knew?
• Could you solve it another way?
• Which way felt easiest?
• Can you show that strategy with a Montessori material?
• Can you now solve it without the material?

This does not mean we need to turn every maths lesson into a long discussion. Sometimes a quick question is enough to reveal whether a child is counting everything, using place value, relying on a known fact or applying a more efficient strategy.

We can then decide whether the child needs:

• more experience with the concrete material;
• practice building number knowledge;
• a presentation of a new strategy;
• help connecting two materials;
• opportunities to apply the same idea in a new context;
• or encouragement to move towards abstraction.

Children do not need to use every strategy

Just as a child does not need to complete every Montessori material, they do not need to use every possible calculation strategy.

The aim is not to teach a long list of compulsory tricks. The aim is to give children a flexible understanding of number so they can choose an approach that is accurate, efficient and makes sense to them.

Different children may solve the same equation in different ways. That can be a strength. Comparing strategies helps children see patterns and understand why a method works rather than simply memorising a procedure.

At the same time, we should continue observing efficiency. A child who counts every bead to solve a multiplication fact they already know may need help connecting that known fact to the new situation.

Assessing knowledge and strategies

To help teachers find out what maths knowledge a child has and which strategies they are using, I designed a Montessori maths assessment.

It is not designed to grade children. It is designed to help you identify useful next presentations and lessons. The assessment explains what each question is assessing and provides ideas for what you might teach next.

Explore Montessori Maths Resources

Montessori resources for developing maths strategies

Final thoughts

Montessori materials are powerful because they make mathematical relationships visible. However, the material is not the final destination.

We also want children to recognise number patterns, use facts they already know, transfer their understanding between contexts and gradually develop efficient mental strategies.

By observing carefully, asking children how they reached an answer and helping them connect materials with number relationships, we can support both deep conceptual understanding and increasing mathematical fluency.