Last month, we explored understanding numbers using concrete representations. Identifying physical objects or fingers makes it easy for youngsters to understand quantities. We can then progress them to recognize pictorial representations and eventually to learn to assign the symbols we call numbers.
Numeric symbolism, however, is not meant to replace the understanding of quantities visually. Symbolism instead builds on a foundation of visual and concrete associations with numbers. The more deeply students understand that numbers are quantities they can picture, the more easily they can manipulate those quantities in their minds to perform arithmetic on them.
This leads us to another one of our most important underlying teaching philosophies: VGCC – Visualizing, Grouping, Connection, Calculation.
VGCC: Visualizing, Grouping, Connection, Calculation
Whether it’s for comparison, pattern recognition, or the four basic operations of arithmetic, mathematics is all about moving numbers around. And as any beginner in math knows, moving numbers is made a lot easier by the help of physical or visual aids.
Visualization
That’s because a visual depiction of a number clearly shows how much it’s worth, and therefore it is much easier to see how much it will be worth once it has been manipulated. At Mentorhood, we use Singapore Math techniques because we strongly encourage students to instinctively visualize numbers on a foundational level.
Grouping
From a visual perspective, basic arithmetic involves grouping and moving numbers for the sake of change or comparison. Subtraction separates numbers into what gets taken out and what gets left behind or comparing two numbers to see how large is a group of outliers. Addition involves taking two or more groups of numbers and seeing them as one. Of course, multiplication and division involve forming or breaking down large quantities into smaller groups.
Connection & Calculation
Students who can visualize numbers in their minds can see what is happening to these numbers as they perform operations on them. They can form a connection between the requirements of an operation and the numbers they are working with. Visualization will not only help them to pick up the functions of operations more easily, but they will also be able to translate this understanding into quicker and more confident calculation and problem solving.
How to Teach the Power of Grouping
We like to constantly reinforce familiarity with basic relationships between numbers. When students start working with addition and subtraction, we recommend having them practice making groups of ten, or taking away from ten. When they start to work with two-digit numbers, add another group of ten so that they can see what patterns continue as they try to make twenty.
The easiest way to start working with grouping in single-digit scenarios is to use one’s fingers. Using fingers makes it easier to teach students to recognize groups of numbers that fit together to make ten. As they move into two-digit numbers up to twenty, they can apply this skill further.
For example, once they learn that 3 + 7 makes a group of ten, then we can teach them to connect that starting at 13 and adding 7 would get us to the next group of ten, which would be twenty.
As they begin to get comfortable with the ideas that similar groups of numbers can have similar results, we can start to increase the numbers of digits and work with simple numbers. Groups of 100 are a lot easier to add up than groups of 99, but if students can visualize putting the simpler groups together, we can help them make the connection between that problem and a more difficult one.
Similarly, grouping is a powerful tool when it comes to problem solving multi-step questions. As you work with your student on a word problem, encourage them to visualize the groups of numbers they read about in the problem. This will help them to keep track of the information they gather as they start to solve the question and see the gaps they need to fill.
Grouping it All Together
Grouping is about understanding parts of a whole and wholes that are made up of several parts. These relationships are foundational for a layer of operational calculations on top. Grouping allows students to practice isolating variables, separating and analyzing numbers, and comparing quantities.
Foundational to this is the ability to visualize numbers so that students can move them around in their minds. And this skill is built on a comfort with the concept that numbers represent tangible quantities. As students build on their most basic introduction to numbers using physical objects, they create a portal from the physical world into the abstract world of problem solving in a mathematical space.