Making Math Touchable

Making Math Touchable 

When I was in elementary school, the only people who used manipulatives were kids in grade 1 or 2 who were weak in math and needed extra help to understand basic concepts. These students were given manipulatives because they couldn't understand otherwise. Students who were smart didn't use manipulatives, because they didn't need them to understand the concepts.

After playing around with a set of double sided counters, I have discovered that I like manipulatives. While I don't feel like I require them in order to understand math, I do feel like I understand the math better while using them. Before I started playing with them, I didn't think that there was much of a difference between using manipulatives and images, but using the manipulatives puts more focus on the transition between stages of solving a problem, while images tend to simply show the stages themselves. I remember being in school and understanding that the teacher had went from Point A to Point B, but I didn't understand what had happened in the middle. By using something that I can physically move around myself, I would be able to understand and remember what is happening better in that situation.

I can understand why many teachers have avoided using manipulatives in the past. From a classroom management perspective, they create more ways for students to be distracted and to cause a mess in the classroom. Another concern is that students become reliant on using these concrete aids, and never progress to understanding the concept abstractly so that they can simply use mental math in the future. Teachers should always stress to their students that the manipulatives are simply an aid to learning, and that they are simply one way of many to view a particular concept. Manipulatives are a great way to explain a new concept, but afterwards the teacher should transition into having the students practice the same concept without the manipulatives so that they understand it also abstractly.

This is an example of one way to use counters while teaching integers to explain the concept of adding negative numbers, and how it is basically the same thing as subtracting a positive number. The red counters represent negative numbers, and the yellow ones positive numbers.

We begin with 5 red counters. 

Equation: -5

We are told that we need to remove 7 yellow counters.

Since we don't have any yellow counters, we need to find a way to add them into our equation, while making sure that we don't change our overall value from still equaling 5. We can do this by adding "zero pairs", which are equal amounts of both positive and negative numbers. Since we need to later remove 7 yellow counters, we will use 7 zero pairs consisting of 7 yellow counters and 7 red counters. 

Equation: -5 + (-7 + +7) = -5  

Since we now have 7 yellow counters in our equation, we can finally take 7 yellow counters away. 

We are now left with our original five counters, and addition the 7 red counters that we needed to add in order to remove the 7 yellow counters. Since we needed to remove yellow counters but not any red counters, we keep the red counters that we needed to use to add in yellow counters to be able to remove the yellow counters. 

Equation: -5 + (-7 + +7) - +7 = -12

Simplified equation: -5 - +7 = -12 

Simplified equation: -5 + -7 = -12 

Making Math Memorable 

While it is critically important to teach the logic behind math, and how it rationally works, it is also practically important to teach little tips and tricks to help students solve questions quicker. Many students will create their own formulas or shortcuts after learning the logic behind a mathematical concept, but other students will struggle with this, and will find themselves working through the entire problem solving process each and every time. For these students who struggle with math, doing practice math can feel overwhelming because it's so time consuming, so it's helpful for them to be taught shortcuts or formulas after they've learned a certain concept to help them answer future similar questions more quickly. Teachers do need to be careful that these students aren't relying on this shortcuts entirely, but are using them as tools for specific situations.

Besides formulas that make problem solving quicker, there are other tricks that can help students remember facts. There are many little rules in math that are easy to confuse or forget inside of a larger problem, despite how logical it looks when you isolate the mathematical concept.

One example of a memory trick that I learned this week was this little chart that explains how positive and negative numbers interact together when they are added or subtracted from each other:

There is a story that goes with this chart: "When a good man comes to town, the townsfolk are happy. When he leaves, they are sad. When a bad man comes to town, the townsfolk are sad. When he leaves, they are happy."