Modifying Math Misconceptions

It is impossible to teach someone when they believe that they have nothing left to learn.

When I was a student, when I believed that I understood a mathematical concept, I would zone out in class as the teacher repeated herself for the third time, ignore the in-class activities, and then not do the homework questions. From my perspective, why should I waste my mental energy focusing on something that I don't need further clarification on when I could do something fun like doodling a new design? When we had quizzes or tests I discovered if I had actually understood the concept like I had thought, or not. More often than not I understood the concept enough to get around three-quarter of the questions right. Since I was doing well enough that I was passing the course with decent marks, I didn't care about the questions that I had gotten wrong, or if I hadn't actually understood the concept quite properly or completely.

One of the most important things that we need to accomplish as teachers is to engage our students in the lessons that we are teaching. I have always been interested in literature, and so I always paid attention in English class, even when we were reading novels that I didn't really enjoy. While I was in university, one of my professors showed us a certain TED talk, and I have re-watched this particular TED talk several times since. On YouTube I have watched informational videos on topics that I am already educated on but that I feel very strongly about just to see someone else's specific views on the topic. I have mentioned all of these examples to show that a student who is interested in a topic is much more likely to remain engaged when they already feel that they fully understand the topic.

Many students believe that it they understand a topic "well enough" that that is "good enough," because they care more that they pass the course with a reasonably good grade than that they completely understand a certain concept. This attitude is why so many students develop and maintain misconceptions about mathematical concepts. When these misconceptions are carried over from previous years, they can significantly affect how well that student succeeds when that mathematical concept is expanded upon in an older grade. The student can become confused when the teacher is teaching the lesson, because they are experiencing dissonance between their own understanding and what the teacher is saying. Often when these students ask for clarification, the teacher struggles to find a way to make the student understand when they don't address the misconception that is holding the student back. Other students will assume understanding of a topic, and if they are not engaged in the lesson, won't pay attention and so learn that they are wrong. When these students do badly on tests or quizzes, their low marks discourage them and cause them to dislike math.

One method to engage students while also correcting misconceptions is to directly confront common misconceptions that our students likely have.

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." 

Making Math Entertaining

Just how does one make learning fractions fun?

Should the teacher talk in a funny voice while teaching the lesson? Are neon coloured numbers enough to add in excitement?

While these examples are hyperbolic, they do represent teachers who try, but ultimately fail, to make math interesting. The problem with this type of teaching is the teachers try to add something "fun" on top of their lessons to make them engaging, but the lessons themselves remain the same. In order to make teaching truly engaging, teachers need to dig down deeper beneath the surface level of their lessons, and change the lessons themselves.

This past week, I had fun while doing math, which was an interesting experience for me. Our teacher gave us a tarsia puzzle to solve during class, and I felt disappointed that class was over before we finished the puzzle.

Our tarsia puzzle in progress

The tarsia puzzle reminded me of playing dominoes. Each participant is given a set of triangles, and they must work with their teammates to match up each of the triangles with its correct partner. Unlike dominoes, where there are multiple ways to build the pattern, with a tarsia puzzle, there is only one correct pattern. The real challenge of the puzzle is finding not just a pairing, but the particular pairing that works to make the entire final shape work. My classmates and I had put together quite a few pairings before we realized that they weren't creating the correct shape, and so we had to rethink the entire puzzle.

A finished tarsia puzzle - colleenyoung.wordpress.com

What I really enjoyed about the tarsia puzzle was that it took my focus away from the math that I was learning, and gave me a goal that I actually enjoyed working towards. One of the problems that I personally have with math class is the fact that it's so boring and seemingly pointless. I generally find it extremely difficult to sit and practice creating a list of equivalent fractions, yet, while I was working on the tarsia puzzle, I was doing exactly just that with ease.

Another method of teaching fractions in an actually interesting manner is to make them meaningful in a practical setting, for example making smoothies. This site teaches students about fractions while teaching them how to make smoothies. The thing that I like about this site is the same thing that I liked about the tarsia puzzle - how they both shift the focus of the activity away from the math in the activity, and onto something else. By making math not the end goal in and of itself, but rather a means to a goal, makes it feel a lot less tedious.