How Do We Know When Elementary Students are Ready for Abstraction?

by Letty Rising

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One of the most inviting aspects of the Montessori approach is our beautiful materials.  Other than the idea of following the child, the enticing materials are one of the main reasons why people are drawn to Montessori environments. And, how could anyone refuse these beautiful, sometimes delicate, always carefully crafted materials?  

They are a delight to the senses and appeal to many people who look back on their own educational journey and want something different for children.  Something other than what most of us had:  sitting in rows of desks with the teacher as the “sage on the stage,”  using pencil and paper to compute math problems that we often didn’t fully understand, but hoped to “get right” by following a formula or sequence.  In truth, I didn’t understand the concept that a square root was just the amount of one side of a square until I took my Montessori training, where my mind was blown wide open! 

When to move to abstraction?

So let’s say you’re humming along as a first-year teacher, or maybe even a second or third-year teacher or beyond.  You’ve mastered your lessons, you have presented them, and the children are repeating the work.  But one day, you notice that many of the children who were working on the checkerboard, for example, have been using it for a long while, and it’s dawned on you that most traditional students would be tackling this work on pencil and paper by now, and  that you’re not quite sure when to introduce paper and pencil.  You scour your albums and don’t see anywhere that indicates after how long students use the materials that they should move on to abstraction.  

Like many aspects encountered in the Montessori approach, the response to such dilemmas can often be “It depends.”  Passage to abstraction can happen after a child has worked with a material for a couple of days, a week, for several weeks, or even for months. The younger the elementary child, the more likely they will be spending a longer amount of time with the materials. 

However, this doesn’t mean that, because upper elementary children can arrive at abstraction more quickly, that you want to rush aside or even brush off the materials!  Older children find squaring, cubing, non-decimal bases, and powers of numbers, just to name a few, to be endlessly fascinating with the materials.

While some people might think of squaring to be a material designed for older children, the earlier squaring lessons are perfect for children around ages 8-9, as it offers an additional opportunity to repeat multiplication and aid in memorization of multiplication facts. What would be more fun, practicing with squaring material and/or using graph paper to create colorful squares, or filling out a multiplication worksheet? No contest! 

Signs a child is ready to abstract a concept:

The child is consistently completing problems using materials, with a great deal of accuracy. How much is a great deal, you say? About an 85 percent accuracy or better, and the ability to apply provided strategies to overcome consistent errors. For example, it might be that the child understands the work conceptually, but they are making minor computational errors. Showing them how to slow down and check their work when finished, and also seeing them demonstrate “backtracking” when a mistake has been made and being able to correct it, while getting most other problems correct, is a good sign that they can confidently abstract.

The child has been repeating the concept on materials for some time and have expressed or demonstrated boredom.  It is important to note that, while the child might be bored with the material, this may not be a sign that they are ready for abstraction, but that they are tired of doing the same thing over and over again. This is why we have repetition through variety. 

For example, children learning long multiplication have repeated experience with the large bead frame, the checkerboard, the flat bead frame, and the elementary bank game.  Maybe a child has grown weary of the large bead frame or the checkerboard…have you shown them the flat bead frame?  There is no need for the children to use every single material that involves multiplication…children often have their “favorite” and they might be ready for abstraction after using their favorite for a period of time. Or, they may be ready for a different material, to give them that “repetition through variety” experience.  As always, observe your students, and talk with them. They will let you know what they need. 

Child expresses an interest in working out problems abstractly. If the child asks to compute problems on pencil and paper, let them try! You will want to model this, which is explained in further detail below. If they are asking to work abstractly, then try it. The worst that can happen is that the child needs more time with the materials, and they will experience firsthand, by their initial stab at abstraction, that they need to work hands-on a bit more. 

A step between concrete and abstract

Some of the math lessons offer that step between concrete and abstract…using graph paper to color the geometric form of multiplication, for example.  Also, squaring using graph paper is a delightful activity that serves as a bridge between the squaring material and paper and pencil equations.  

Is it okay that children are working abstractly with one math concept while still working with the materials in other concepts? Yes! In fact, please resist the mindset of “I can’t move forward with teaching Johnny new lessons until he’s mastered addition with the stamp game.” This is an old paradigm way of thinking about math, that a child has to perfect one area before moving on to another, and in fact, can kill motivation.  Different strands of math are often presented simultaneously, and while you want to maintain the sequence within a strand (e.g. the child needs to master division with a one-digit divisor before moving onto a two-digit divisor), students can simultaneously be working on different math topics, and at varying levels of abstraction.  

Abstraction: How to demonstrate

Teach them how to abstract by matching the paper process to the corresponding material. For example, if you are helping them abstract for addition, you can pull out the stamp game, and a piece of paper, and every step you do with the stamp game, you then record on paper. When you are exchanging, you say something like “here I have 16 units, remember that whenever I have 10, I exchange those units for one ten. And now on my paper, I’m going to write this “one” on top of my tens column, which is the same thing as me adding another “ten” to the stamp game.”  Talk through every step as you align the material to the algorithm.  

Important points to consider when moving from the concrete materials to abstract algorithms:

• Some children will move quickly into abstraction, and then return to the materials for advanced applications of previously mastered or abstracted concepts.
• Understanding exchanges requires working with a wide range of materials.  Elementary children can find working with the stamp game to become quite tedious, and they might be craving abstraction but not yet ready for it.  The Dot Game, often introduced in a 3-6 classroom but works quite well for the younger elementary child, can be a helpful bridge towards abstraction. 
• When moving to abstraction, sometimes a whiteboard and markers is a wonderful first step before using pencil and paper. I’m not sure if it’s the ability to easily erase, but children can hardly resist using whiteboards and markers!

Abstraction: the end goal

It is important to remember that the materials serve the child, and not the other way around.  Maria Montessori designed several math materials to demonstrate a single concept because the elementary child has lost the desire to repeat something to the point of perfection. The materials give us an opportunity to say “let me show you something new” (while still demonstrating the same concept).

And while these materials are beautiful, and they are useful, and they are helpful, the most important thing to remember is that they are an aid to the child’s development. They are the starting point, and not the final destination. We want elementary children to move through the materials with interest and delight, and also to leave the classroom with the capability to work out math problems abstractly with a conceptual understanding that was developed through experiential practice with the Montessori math materials.