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.

Math as a Three Legged Stool

Memecenter

When I was in highschool math class, I generally swung between either thinking that mathematical concepts were extremely easy, or they were extremely difficult. So often I would follow along with the examples that my teacher wrote on the blackboard, understanding everything (or so I thought), correctly answer the first few homework questions, not do the rest of the homework questions (I clearly understood the concept because I answered the first ones correctly, and I had better things to do with my evenings), and then wonder why I got so many questions wrong when I received my unit test back. This repeated experience served to make me increasingly frustrated with math, and continuously doubt my math ability while doing nearly anything. My problem was that I was so focused on getting through my homework questions as fast as I could that I would use a single formula or process to answer every question, without thinking about whether there was anything else that I would need to do. I had zero adaptability when it came to answering a different type of question based on the same basic concept. Mind you, my teachers only ever taught by formulas and examples, so they did not shine as beacons of adaptability to me or my fellow classmates.

Looking back, I do believe that our teachers assumed that we would gather all of the information that we needed from their simpler examples to answer the more complicated questions. Whenever we had questions in class, they would direct us to review the example written on the board, or would provide yet another example following the same formula or procedure as before. They were effectively training me in procedural understanding (how to follow step by step instructions), while assuming that I would pick up conceptual understanding (the how and why of the concept) without ever explicitly teaching conceptual understanding. I was so used to procedural understanding training, that I specifically would request this from my math teachers, believing that the only way that I could do the math was if I had a formula handed to me.

As an adult, things have flipped for me. I have lost nearly all of the procedural understanding that I used to have, but instead I work through every problem using my conceptual understanding. This is a very slow process for me, since I have to work my way through each individual step before I can put all of the pieces together.

Neither of these situations are ideal.

Edugains has a fantastic poster resource that shows how instrumental (procedural) and relational (conceptual) understandings are both important to math education, and how they affect each other. As the poster states:

"Life-long learners of mathematics build new knowledge and skills in prior knowledge using the mathematical processes 

Confidence and flexibility results from conceptual understanding 

Accuracy and efficiency results from knowing facts and procedures" 

Some people like to argue about procedural versus conceptual, like they are two mountains separated by a deep valley. I do not see them as separate entities though, but rather as the two sides of a coin. As an adult, I do tend to place more importance on relational understanding, but this is more so due to a worry that it isn't being included in classrooms than that I devalue procedural understanding. I believe that when a person has a strong relational understanding, they develop procedural understanding naturally based on noticing repeated patterns while they engage in problem solving.

Procedural understanding, conceptual understanding, and problem solving are the three legs that support all of mathematical understanding.

My Reliance on Formulas

When I was a student, a large part of the reason why I didn't like math was because I viewed it as a set of very specific skills. Either you knew how to find the volume of a sphere, or you didn't. And if all that math was were such specific skills such as calculating the volume of a sphere and the like, and you didn't have a specific reason to use such particular skills, then math was useless. Looking back now, I do believe that I held this perspective because of the way that I was taught math in schools as a very formulaic process. Every single class followed the same pattern: my teachers would show us a particular formula at the beginning of class related to the specific area of math that we were learning about, do multiple examples on the backboard following this precise formula, and then make us practice the formula by answering the questions in the textbook. There was no explicit mention of the seven part mathematical process referenced in the Ontario Math Curriculum, nor was there focus placed on problem solving except for figuring out how to adapt a different problem to fit into the formula that we were taught.

"An information- and technology-based society requires individuals who are able to think critically about complex issues, analyse and adapt to new situations, solve problems of various kinds, and communicate their thinking effectively." - Ontario Math Curriculum, page 4

Moving forward as a teacher, I will not be imitating the teachers who I had when I was a student. Instead, I am planning to follow the guidelines in the Ontario Math Curriculum about how to teach problem solving skills. The Curriculum recognizes that there is a general negative attitude towards math in our society. To counter this negativity, the Curriculum advises teachers to value various problem solving approaches, stating "Students need to understand that, for some mathematics problems, there may be several ways to arrive at the correct answer" (page 26). This is a direct contrast to the single formula that I was taught. I am also impressed with the way that the Curriculum is promoting cross-curricular and integrated learning, where math is taught together with another subject like science.

 I want my students to see math as a much wider set of skills than a list of specific formulas. I am considering using a journal approach with my students where they will have to reflect on the process that they used to solve a problem so that they become aware of the skills that they are developing as they work on solving these problems. I also plan to teach with cross-curricular lesson plans so that my students see the relevance of the specific skills that they are learning in a real life context.

However wonderful my pedagogical ideals are, they alone are not going to turn me into a fantastic math teacher. I am going to have to work on my content knowledge. If I want to teach my students to understand math logic as they develop their problem solving skills, I myself am going to have to understand this math logic myself.

"Teacher knowledge makes a substantial contribution to student achievement" - Toward a Practice-Based Theory of Mathematical Knowledge for Teaching, page 4

I believe that a good math teacher needs to have two types of knowledge: content knowledge, or an understanding of how math works, and pedagogical knowledge, or an understanding of how best to teach mathematical concepts. I felt that most of my teachers had good content knowledge, but needed to improve their pedagogical knowledge. I am concerned that while I will have fairly good pedagogical knowledge, that I will have a fairly terrible content knowledge. Reading a quote from Toward a Practice-Based Theory of Mathematical Knowledge for Teaching made me reflect on the state of my content knowledge. The article stated, "Second, looking at teaching as mathematical work highlights some essential features of knowing mathematics for teaching. One such feature is that mathematical knowledge needs to be unpacked. This may be a distinctive feature of knowledge for teaching. Consider, in contrast, that a powerful characteristic of mathematics is its capacity to compress information into abstract and highly usable forms" (page 11). The formulas that I learned as a child were not problematic in and of themselves, because they are the compressed form of math referenced in the article. The problem is when the formulas become the end-all and be-all, and aren't
ever unpacked to explain the math logic that created the formula. I fear that unless I develop a better understanding of mathematical concepts and logic, that I am going to slip into teaching only formulas because that is all that I understand myself.