How to go From Concrete Manipulatives to Abstract Math
Updated: Apr 17, 2022
As educators, and homeschool parents, we know the importance of using concrete, hands-on materials, or manipulatives, to help children gain a deep understanding of math. We also know that as adults, we don't pull out manipulatives when we need to calculate math in the real world. So how do we help children move from the concrete materials to the abstract representation on paper? We don't want to move to fast, and risk blunting their understaning, but we also don't want to keep them dependent on tools that they no longer need. Dr. Montessori's advice was to, "follow the child."
In this post, I will tell you when to use math manipulatives in lessons, and how to know when children are ready to go on without them and work in the abstract.
1 - Introduce Manipulatives
The purpose of concrete materials is to build understanding. So, when teaching children new math concepts, always start with concrete materials! Very young children learn through using their hands, and absorbing impressions from their environment. Numbers or even drawings written on paper, or a whiteboard are abstract symbols that don't have the same meaning to them as they do to an older person. Even as children get older and develop the ability to think abstractly, they still need a concrete impression to build their abstract ideas upon. So to introduce a new math concept to a child, always start by showing them how to use the hands on material. The lesson should be short, and introduce only one topic at a time, isolating the difficulty.
2 - Use Few Words
One common mistake adults make when teaching children math is trying to explain a concept with words. This often confuses children, and distracts from the hands on material. Well designed math manipulatives, such as the Montessori material, will embody a concept, and enable the child to easily discover, figure out, or see the concept just through use. Therefore, when presenting a lesson with hands on tools, it is critical that the teacher use as few words a possible in the lesson to allow the child's attention to remain on the materials.
"The child doesn`t learn by listening to explanations but only deepens knowledge through active work and frequent, prolonged and patient practice with the same thing (which has been understood)"... Dr. Maria Montessori
3 - Hands On
After a concept has been introduced with hands on materials, it is critical that the child is then allowed to try it out and practice repeating the activity as much as they like. A common mistake I often see educators make is not allowing the child to practice with the hands on matierial enough, or at all. Young children love repetition, and will naturally want to repeat the same activity over and over. Older children need more novelty, and so educators must find a variety of ways to allow for practice of the same concept with the hands on tools. The child should be allowed to repeat the activity until they feel satisfied that they understand the concept.
4 - Record Work
Depending on the lesson and the child, it can be helpful to first only show the concept, and not expect a child to write anything down. This can allow the child to maintain focus on the material, and removes the secondary difficulty of needing to also write while learning something new. This is especially helpful for children with learning differences or fine motor challenges, as it makes the learning process less stressful, and more joyful. Once the child understands how to use the material, recording work can be shown as a next step. CLICK HERE for my Montessori printables bundle.
5 - Allow Repetition
It has been said that it takes three to twenty times to learn something new. Children need lots of opportunities to practice witht the concrete materials, and may sometimes need lessons to be repeated. Once they remember how to use the tools though, they need lost of independent practice and repetition. I like to use task cards for this purpose, with only one problem on it at first. This eliminates the business of worksheets, and allows children to do work in small chunks. As a bonus, there is less for the teacher to copy, as the cards can be laminated and reused. I once had a student who would shut down whenever I put a worksheet in front of him, but lit up when I cut out the problems into little squares, and only showed him one at a time. I have created my own math task cards that I use in my classroom instead of worksheets. I keep them in small containers for children to freely access as independent practice with the hands on material. CLICK HERE to get my addition, subtraction, multiplication, and division task cards.
6 - "I Figured it Out!"
The beauty of allowing children to practice a math concept for as long as they need to using hands on tools, is that eventually, many of them will discover a rule, pattern, or develop strategies themselves! They may say something like, "I don't need to use the hands on tools anymore, because I figured it out, and can do the math in my head now!" This can be and extremely joyful moment for a child. It generates the excitement of discovery as opposed to the boredom and confusion of listening to the teacher trying to tell the rule, pattern or strategy to a child. It is not the time to put away the concrete material just yet, however!
7 - Prove It!
Once work with concrete material has led the child to a deep understanding of a math concept, you can tell the child to show you what they discovered! You will be able to see if they actully are able to do the math. If the child can indeed show you they can do the math on their own without the material, have them try some more problems in their head, and use the concrete materials to "prove it". This shouldn't be a forced step, but an opportunity for a child to prove to themselves that their discovery works in all cases! This is a key moment, and a clue for you as the educator that they are ready for the next step!
8 - What is the Rule?
At this point, you might ask the child to explain to you how they work a problem in their head. If it applies, they can explain to you the "rule" they discovered. For example, they might say, "To divide a fraction by a whole number, you just divide the numerator by the whole number. The denominator stays the same!" Or for a division problem, they might say they make it a multiplication problem in their head and the missing factor is the quotient! If the child doesn't discover the rule or pattern on their own, of course you just tell them. But doing it this way is much more meaningful than having them learn from an abstract vocabulary card at the begining of a lesson or chapter!
9 - Bridge to Abstraction
With many of the Montessori hands on math materials, the child is able to make amazing discoreries about the patterns and rules behind mathematical concpets. However, some concepts need lots of practice before full abstract understanging is achieved. There are materials in Montessori designed to be a bridge to abstraction. These materials spiral back to concepts, give children extra practice in a novel way, and help lead them to efficiently do their math. A good example of this would be the addition strip board and finger charts. When learning the addition tables the Montessori way, children start with the concrete: putting together two number rods to match the size of another. For example, the two rod and the eight rod are the same size as the ten rod. Next, children add with the colored bead bars, and find the answers by counting individual beads. For example, the two green beads strung together plus the eight brown beads strung together add up to ten beads in all.
When they have had plenty of practice with the concrete material, they are then introduced to the addition strip board and finger charts. The strip board is made up of red and blue "rulers" made out of card stock paper, cardboard, or plastic. Each ruler is made up of a certain number of squares, and labeled with that number. For example, the child lines up the blue ruler with two squares and the number two written on it, and the red ruler with the number eight written on it, and see that there are ten squares! This bridge to abstraction also helps the child to learn to count on. CLICK HERE for a printable version of the Montessori addition strip board.
The finger charts are used after the child understands what and how to add. This allows them to quickly look up an addition fact when they have had plenty of practice with the rods, and beads. CLICK HERE for a printable addition finger chard.
10 - Abstraction
If a child has shown that they understand a math concept, and have shown you they can now solve problems mentally, there is no need for them to continue using the concrete materials to complete their work. There is also no need to require a child to make a tedious drawing to represent the hands on tools. This does not mean they will never need to use manipulatives again, but just not for the particular concept in question.
Concrete to Abstract
Knowing how long you need to have children use hands on math tools comes down to following the child. Taking tools away too soon, or forcing their use longer than needed can both have negative affects. Remember, the purpose of the tools is to help the child develop their own understanding of a math concdept. Follow their lead, and you will be fine! CLICK HERE for more printable math resources to help your kiddos work from concrete to abstract math.