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Mathematics Awareness Month: Week 2


April is Mathematics Awareness Month: Week 2

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Algebra

If this is a mathematical word that brings back some unpleasant memories, this week's newsletter is for you. There is a growing body of research evidence supporting the idea that teaching algebraic thinking in elementary grade classrooms is not only the correct approach, it's also a great way to create the foundation for future success in mathematics.

So, what does this mean for our classrooms and your child?  Perhaps more importantly, what would this look like in your child's math lessons? It's important to immediately note that an early introduction to algebra is not about teaching algebra using the methods with which we were taught. What we are going to examine here is how we can support the development of ways of thinking, doing, and communicating about mathematics.

Our instruction should be focused on teaching mathematics in ways that result in children understanding mathematical concepts and developing mathematical insight. We need to focus on providing opportunities for children to experientially recognize connections, analyze relationships, making conjectures, justify their mathematical thinking, and notice structures. In other words, we will focus on building the foundational habits of mind that will prepare children for their future learning of mathematics. It's an exciting venture!

This kind of instruction has to include experiences that allow students to learn through activity and discussion, not through simple memorization of facts or the steps of an algorithm. I have mentioned Jo Boaler, a Stanford University professor of mathematics in previous newsletters. In a Hechinger Report commentary, Boaler wrote that "we continue to value the faster memorizers over those who think slowly, deeply, and creatively," and that this has "produced a generation of students who are procedurally competent but cannot think their way out of a box."

I remember, from my high school algebra classes, being introduced, for the first time, to the Commutative Property, the Associative Property, and the Distributive Property.

Please do not stop reading!  I will explain.  😄


Commutative property of Addition:  2 + 3  =  3 + 2

Commutative Property of Multiplication:  3 X 4  =  4 X 3

In other words, order does not matter, either with addends or with factors. 


Associative Property of Addition:   (1 + 6) + 7 = 1 + (6 + 7)

Associative Property of Multiplication:   (5 X 8) X 2 = 5 X (8 X 2)

In other words, you can group addends or factors in different ways without affecting the resulting total.


Distributive Property of Multiplication over Addition is a bit harder to define, so I will refer you to the Math Warehouse for this:

"The distributive property lets you multiply a sum by multiplying each addend separately and then add the products."

It’s much easier to show examples:

                         7 X (10 + 3) = (7 X 10) + (7 X 3) = 91

                  Or    7 X 13 = (7 X 10) + (7 X 3) = 70 + 21 = 91

 Click here for the Khan Academy’s explanation, providing explicit examples.


My point is this: I was introduced to these ideas and terms as a freshman in high school! In kindergarten through fifth grade, we are now discussing real life examples of what these properties formally name.

In kindergarten, we use manipulatives such as blocks, beans, or marbles, to show patterns of the value of 5. If one child sees an arrangement with 2 on top, and 3 underneath, another child will discover that 3 on top and 2 underneath also totals 5. Others may find several other arrangements of concrete materials. The class might then create a poster showing all their findings. Students would develop an understanding that the value of 5 can be represent in several different ways, and all ways depict the amount or value of 5.

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What becomes most important is that students construct these models with concrete materials, rather than by using pencil, paper, and symbols.

The symbol for five is the digit "5." The value of 5 (or 5-ness of 5) is:
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This is why children can memorize number names to one hundred, yet they have little to no number sense with regard to the values of these numbers! A student can sound fluent in counting; what is misleading however, is that rote counting, while essential, does not necessarily mean that the child understands what the number names represent.

Memorizing times tables can result in a similar lack of understanding. Fluency with multiplication facts is desirable, in fact, quick recall of facts is essential for efficiency with calculating and with problem-solving. How children develop their fluency is what influences their proficiency with advanced mathematics. If they rely completely on memorizing tables, they become vulnerable to a potentially faulty memory. Panic can ensue, and the brain’s ability to think of alternative approaches to find answers becomes unavailable.

A more preferable way to think about finding multiplication answers is to apply the Distributive Property to a problem and solve it quickly through mental math.

Confronted with 7 X 13, one can think about breaking 13 down into more manageable addends. 13 can be thought of as 10 + 3.
From there a student can think of (7 X 10) and (7 x 3), and then add the products to determine that: 70 + 21 = 91.

Introducing and practicing algebraic thinking in our mathematics lessons through hands-on experiences will result in students developing flexibility in their thinking and in their approaches to calculating and problem-solving. They will demonstrate an ability to use a variety of strategies to find solutions and successfully complete algorithms. Perhaps more significantly, they will approach mathematics lessons with confidence and enthusiasm, because they will believe they will understand the mathematical concepts they are expected to learn.