Math: to Hate, or not to Hate?

Math is a popular subject to hate. After all, who likes to be forced to sit in a stark classroom, attempting to answer an extremely long list of questions that all look like gibberish for an hour straight? Or who wants to tell their son or daughter that they have to endure this situation? Regardless of whether this is what the modern math classroom looks like, this is what many people associate with math. Their own personal experiences are combined with the attitudes presented in numerous Hollywood movies, and these together mix to create the negative atmosphere that surrounds math. Before they even begin school, many students are receiving messages that math is difficult, boring, tedious, and pointless. Within this atmosphere, students begin to internalize certain beliefs about math, and about themselves, such as that math is boring and useless, and that they are either good at math or bad at math. Personally, I believed that math followed a strict set of rules, and that there was essentially only one way to solve a mathematical problem. I also failed to see where I would use any of the abstract equations that I was attempting to solve.

I was taught to solve mathematical problems using formulas. These formulas were handed to me as pre-made tools from my teachers, and I had to plug in the correct numbers. This style of teaching did not foster my problem solving skills, not did it inspire any curiosity that would lead me to make new connections between the facts that I had already learned. It was very difficult for me to transfer my knowledge to a similar problem if it did not follow the exact formula that I had been taught, because I didn't have the background understanding of how the formula had been created. Between my lack of interest and struggle to truly understand the concepts being taught, I gave up investing anything more than the minimal effort required for a decent grade.

However, while math is born out of logical reasoning, it is nothing like the rigid structure that I viewed through the narrow lens of formulas, and there are multiple ways to come to the same answer. Many of my teachers taught from the perspective that students must be able to solve questions, but their approach overlooked the importance of us learning the very basic structure of math itself. While they taught us what to do, we didn't understand why we had to do it this way. Teaching through inquiry places the focus on understanding this structure instead, and students naturally find their own ways to the correct answers by exploring mathematical relationships using tools like manipulatives. To quote mathematician Georg Cantor: "In mathematics the art of proposing a question must be held of higher value than solving it." Finding specific answers isn't as important as figuring out a way to find the answers. Students who learn the structure of math through inquiry and exploration are developing problem solving skills that will enable them as they explore increasingly complex concepts.

Mathematical Measurements

How confident are you when you make guesses about someone's height?

This was a question that one of my teachers asked our class before, and our class had a range of answers from "hardly" to "pretty confident". This question was a precursor to an experiment where we had several student volunteers come to the front of the classroom, and have everyone guess their height in centimeters, then afterwards compared our guesses to their actual height. The results were interesting, with a range of accurate guesses and wildly wrong guesses. What fascinated me the most was the fact that while making their guesses several of my classmates had wrongly assumed that certain volunteers were taller than others, which completely skewed their guesses as to how tall the volunteers were.

When we did this experiment to test our observational skills, I began to think about how important accurate observational skills are, and how I had totally overlooked them before. For example, eyewitnesses are vitally important in our justice system, and they need to provide the police details about suspects so that the police will be able to take the correct person into custody. If an eyewitness gives the police a very inaccurate estimate about the height of the suspect, suggesting that the suspect was average height when he was really above average height, this misinformation can hamper the search for the suspect, or cause them to take the wrong person into custody.

I do not recall learning how to make accurate estimates about lengths when I was in elementary school. I remember learning about the importance of specific measurements, and measuring something correctly, but not how to judge lengths from a distance. As an adult, I wish that I had learned this skill when I was a child.

In order to teach students how to accurately estimate lengths, it's important to teach them several benchmarks to use to base their estimates off of. In Casey's video, she suggests having the students use their hands as one such benchmark for future estimates. One of the things that I really like about this suggestion is the fact that students always have their hands with them while they usually don't have a ruler nearby. One problematic thing about teaching students to use their hands is the fact that elementary students are still growing, and so their hands are also going to grow larger, which will skew their benchmark unless they consciously continue to remeasure their hands regularly to update their benchmark.

There are several ways to teach students about estimation. I really liked the activity on page 36 of this resource that suggested that teachers have students examine several different packaging options for 3D objects, use estimates to make educated guesses about which packaging uses less material, and then measure to test their predictions. I like how this activity goes beyond a 2D concept of distance or length, and ventures into the 3D world, and how it connects this concept with a real world context.

Shaping Mathematical Understanding

There was one single unit in math that actually made me excited for class: geometry. I've always considered this unit to be the artsiest mathematical unit, and as a self-identified artist, I was always drawn to geometry. Working with shapes engaged my inner artist, and satisfied the visual learner in my brain. Naturally, I didn't enjoy every single lesson as much as others, but overall, lessons were at the very least mildly interesting. My favourite lessons were the ones where we got to draw or physically build shapes.

This geometry teaching guide provides ideas and suggestions for activities to teach geometric concepts. One activity that it mentions are paper nets of 2D shapes that can be folded into a 3D shape (page 28). The Ontario curriculum actually has "identify and construct nets of prisms and pyramids" as required information that teachers have to teach (page 82). While I always enjoyed working with nets as an activity in math class, it fascinates me that the Ontario curriculum made nets an actual expectation that teachers are required to cover, because I never realized as a child that other children didn't have the same ease of connecting 2D shapes to 3D objects that I had. This is an area that as a teacher I need to be careful, because I could very easily teach this concept too quickly and assume that my students understand it without properly checking that they do understand, simply because I find this concept so obvious as a child.

Another activity that the guide suggests is drawing a 3D shape based off of a 2D sketch or description (page 29). One of the things that I love about an activity like this is how versatile it is for different types of shapes, and carrying degrees of difficulty, which makes it easy for a teacher to use differentiation in their lesson while using this activity. Not only can the teacher differentiate the complexity of the shape, but they can also differentiate the clues that they give the students about what the final 3D object is supposed to look like.

One of the sub-units in geometry that I found a little bit dull as a child was the concept of location and movement unit. I never had difficulty understanding the concepts that my teachers were describing, but I always asked myself, "Where would I use this?" One year of my teachers connected cartography to this unit, and when I was given a pirate map and asked to figure out how to find to the buried treasure, I found myself interested and understanding the relevance of this information.

Making Math Manageable

My strongest memory of algebra class: seeing and sharing the image above with all of my friends while my teacher rolled his eyes and tried to make us pay attention to real algebra.

Ah, yes, that age old habit of teenagers to disguise their fears with sarcasm. I know I was guilty of this quite often, but hey, it wasn't my fault that math was having relationship problems with its X.

All joking aside, algebra truly is something that many students fear. The word has built up quite the negative reputation for itself, and is often used in popular culture to reference math that is hard, confusing, and unpractical.

However, I don't think that algebra deserves this reputation. As much as I disliked math in school, and found it confusing, personally, I actually didn't have too much of an issue with algebra. It was one of the types of math that I found easier to understand. I think one of the reasons why the letters in the algebraic equations didn't bother me so much was because when I was younger my one of my teachers taught us about the relationship between addition and subtraction by replacing part of an equation with a box. We would need to rearrange the question to be able to figure out the number that was supposed to go in the box. While she used the term "algebra," besides teaching basic numeration skills, she was also preparing us for algebraic thinking. When I started doing math that was labelled as "algebra" and understood that the letters represented unknown variables, I would mentally replace the numbers with boxes, and the question became much clearer to me. Mentally replacing the variables with boxes also put me into the perspective of rearranging an addition question into subtraction, so the process of balancing an equation didn't seem so strange or arbitrary to me.

The one stereotype that I most dislike in popular culture's view of algebra is the conception that it is unpractical. On the contrary, I actually think that this type of math is one that is used on a fairly regular basis. The first example that comes to mind is ordering extra toppings on a pizza. When the base pizza costs a certain amount, and each additional topping costs a certain amount extra, and I'm aiming to keep my final bill under a certain amount, how many extra toppings can I afford? This is mental math that I have actually done numerous times, and so I disagree with the statement that nobody actually uses algebra. I really like the sample activity that the Making Math Meaningful textbook provides on page 620. In this activity, a teacher would ask students to suggest numerous situations where they would use an algebraic expression such as 5N. This activity shows that algebra is more than just sometimes practical, but that it is really a part of everyday living. I also really like the fact that this activity begins with an algebraic expression that is so simple because it works to challenge the perception that algebra is by definition complicated.

Since so many students are so afraid of simply hearing the word "algebra," it can be extremely beneficial to have them working with algebraic expressions, and then applying the label "algebra" to the exercises once they have successfully completed them. Playing games like "Guess the Number" where students have to work through an equation backwards, and creating a single grocery lists for several recipes are great examples of activities where students likely won't realize that they are actually using algebraic skills. Once you can prove to them that algebra is both practical, and that the basic processes are quite simple, you can show them that algebra isn't actually as frightening as they believed it would be, and they will be more likely to be open to learning more.

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.