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The Difficulty of Comparison
3 Popular Manipulatives that Fail to Build Number Sense
Last post, I highlighted all the ways you can build numbers sense fluidly using Cuisenaire Rod staircase work. Many people think that just counting those popular manipulatives all the time is sufficient for building number sense, but they fail to provide depth.
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Those popular manipulatives that will remain nameless (but hold the same design flaws as counting cheerios aka everyday items) are constantly being pushed as the" go-to" manipulatives.
I know. You are disappointed I didn't name any popular manipulatives (yet), but I don't have to. When I show you the superiority of one manipulative over all others, I don't need to name the ones that fail.
It isn't enough to just count objects. The student must grasp how numbers behave and relate to each other.
Students need the big ideas that help them transition with ease from one concept to the next. They need structures that won't break down but carry them all the way from addition to factorials to Pascal's Triangle.
The Difficulty of Comparison
Just look at this picture. It isn't obvious to the child without counting that both groups of cheerios have the same amount.
This forces children to tediously count. This activity is laborious to a child and, need we say, boring.
We don't have to build number sense by counting all the time. It is better for them to subsitize quickly, so that more time can be invested in comparing and contrasting the values and seeing all the diverse relationships that exist between numbers.
Cuisenaire Rods permit a variety of ways to view how numbers relate with more certainty and without all the constant need to count. We saw how staircases provide an overhead view of the relationships that exist between numbers in the previous post. Today, we look at how Cuisenaire Rod Mats provide a more focused view.
Building Numbers
Mat building is a core activity in Gattegno Textbook 1. In the picture below, the student finds all the trains that equal the length of yellow. At the simplest level, this is seeing all the ways you can compose a number. Of course, there is way more going on here.
Compare this to counting just random objects. The student doesn't gain the same sense of how to build numbers without a lot of work on part of the teacher.
The teacher has to create boxes to place the objects inside in order to try to portray some sense of grouping.
Cuisenaire rods allow educators to bypass this and jump right into the concept of number composition. There is no extra work on the teacher's part to demonstrate different ways to group values to build different values. It is all worked out in the design of the rods.
Big Ideas of Math | Properties of Numbers
In exploring each color rod mat, students gain a sense of the properties of numbers. the properties of addition are obvious and intuitive to the student.
They can see that a red plus a purple equals a dark green and that a purple plus a red also equals a dark green.
In fact, they will learn that this principle holds true as they exhaust discovery of the mat. Overtime, they become aware of the distributive, associative and additive properties as well. This terminology will have meaning to the student through the manipulation of the rods.
For a free guide on teaching addition, check this post out here.
It is very hard to grasp this kind of awareness by counting cubes, links and bears. Would I abandon counting objects with a 2 and 3-year-old? No, but I wouldn’t waste my time doing that with a four-year-old when I have a tool that can offer so much more number sense.
Fractions
Cuisenaire Rods are often used to teach fractions. Now, they are excellent for teaching fractions.
However, when you only use them for fractions, you stunt the students perspective on how fractions are related to every other operation from addition to multiplication.
They won’t see everything is a fraction. That fractions are just one way to describe how a number relates to another number. Without the grand perspective of number composition diversity, student's experience tells them that fractions are merely single colored trains. See, how they are missing depth?
Subtraction
Children often struggle with subtraction and mats offer the opportunity to see the connection between subtraction and addition. If they know how to build a number up, they also know how to take one apart.
Cuisenaire rods provide more than one structure to explore subtraction. Staircases offer many opportunities to explore common difference. Common difference is a great strategy to turn an unfamiliar subtraction problem into a familiar problem.
Exhaustive Discovery
Counting beans will never provide a clear, exhaustive discovery of number composition. But why is exhaustive discovery important?
Exhaustive discovery builds character. It challenges students to thoroughly investigate a matter. When students work to find all the ways to build a number, they are faced with the question, "how do I know that I found all the ways to build this number?"
In the beginning, a young student won't explore number composition exhaustively. But mat building provides the opportunity to build stamina to search a matter out. Math isn't just for rapid calculations. It is a playground for thinking minds, and thinking minds need stamina.
Cuisenaire Rods Don't Break Down
Where bears leave you counting, Cuisenaire rods take you far into algebra. Students can explore factorials or Pascal's Triangle.
Clearing out the Clutter with Cuisenaire Rods
Our lives are cluttered with stuff and more stuff. Why have counting bears, cubes and links when you can have something so much better? Why count beans, sticks and buttons when they can't convey the big ideas of mathematics?
I am not saying all you should play with is Cuisenaire rods. We love all sorts of math manipulatives like Anglegs, pattern blocks and 3D building math manipulatives. I just think that some math manipulatives should be tossed out altogether.
