Week 9 - Grade 7 and 8

Being in I/S, most of the courses focus on teaching high school, even though we will be qualified to teach grades 7 and 8 as well. This has led to a lack of preparedness for the case if we end up teaching these grade levels. These grades bridge the gap between elementary and high school, so there is a lot of responsibility to prepare these students properly for high school. In class this week we did an activity where we were given certain specific expectations in grade 9 and were tasked with finding the specific expectations in grade 7 and 8 that would lead to the grade 9 expectations.

This task helped to show us how much math is actually happening in these two years to address the stigma that is around middle school teachers which is that they don't really teach anything in math. When students come into grade 9 they are in a new building with new students in an entirely new atmosphere, so it would make sense for students to initially think that they weren't taught what they had been taught. I remember when I was in grade 9, if a teacher asked us if we had learned a certain concept the year before I wouldn't remember that concept until it was reviewed with us. Once it was reviewed then I was able to remember what we had learned about it, but before that we had all said that we hadn't been taught it. Some teachers, upon hearing that we hadn't been taught the concept, immediately sighed and blamed the middle schools for not preparing us enough for high school, whereas other teachers didn't believe us and went into a slight review to help us remember. I greatly appreciate the teachers that put their faith in our middle schools and didn't believe us because they were the ones that pushed us further into our learning.

Middle school teachers have some of the toughest jobs; they are the ones that help students who are in the process of some major life and social changes to start figuring out who they are and who they want to be, while still helping them learn the required curriculum in order to be prepared for high school. These teachers need the support of high school teachers to better prepare their students for their future. Even if I become a high school teacher and not a middle school teacher, this activity has helped me to better understand my role in the middle school/high school relationship because if I am asked what students need to know for high school, I have a better understanding of how the two curricula interrelate which will help me to give a better response to that question.

Week 7 - Using Tasks

In class this week we talked about how tasks can also demonstrate student learning. An example is the "nana's chocolate milk" video by Dan Meyer. This video uses ratios to solve everyday problems. It shows what he was supposed to do as opposed to what he did do and it asks the viewer to solve how to fix it before continuing on.

Videos like this are great tools for the classroom because the students tend not to realize that they are doing math, compared to a worksheet, and they tend to be more open to solving the problem because of this. Having real life context is essential to learning because it connects the students to what they are learning. This is especially important in a math class where students come in with the opinion that what they are learning will not relate to their regular lives, which leads to the question "why are they learning it?".

However when it comes to context in learning, you can try too hard to make the lesson relative.  Making up an unrealistic situation in order to add context to the lesson is not going to help anyone; it will only make the concept more difficult to relate to. Context has to feel natural in order for it to work they way you want it to.

When I teach a math class I would want to do some research on how other educators were able to add context into their lesson plans and, if the context is realistic, I will incorporate it into my own lesson plans. There are many situations that would add context to math concepts, they just need to be found.

Week 6 - Technology in the Classroom

This week we talked about having technology in today's classrooms and how it can benefit but also hinder education. However, it would only hinder education if the lesson plan didn't evolve with the technology. We aren't teaching the same way that we learned when we were in school because we are able to delve deeper into the understanding of the math concepts through the use of technology. We can ask more open ended questions that cannot be solved directly with the technology but with the technology the students can try out their ideas to see if their answer is correct. This way the technology is not doing the work for the student; the student still needs to know the basic knowledge surrounding the concept in order to solve the problem. Plus, there are multiple answers to open ended questions, so if there are a few students flying through the problems you could ask them to find an additional solution to each problem. We can turn any technology hindrance into beneficial technology by asking the right questions.

Even if the students don't know the correct terminology yet, they can still use technology to start to learn about a math concept. In class this week we used Desmos teacher and Desmos student to play a version of the game guess who. Instead of defining characteristics of the different characters, we had to define different characteristics to determine which parabola the other person has chosen.

We were partnered up randomly through Desmos student and played a game where we each took turns guessing the other person's parabola. This is a fun interaction for a class because they get to use technology, and they increase their knowledge of parabolas. If the students don't yet know the proper terminology of zeros/roots, vertex and so on they can still play this game; they will just need to find different ways to get their point across to the other person. This game also got us to connect with people in the classroom that we wouldn't normally connect with. The first round I was partnered up with two people that I had never talked to before and I found it interesting to connect with them through the game. This game could help the class to connect with each other, even if they wouldn't normally talk to each other inside or outside the classroom.

Technology is a beneficial addition to the classroom and |I believe that it should be used whenever it can because it helps to connect the class, not only to each other but to the learning as well, it helps us to delve deeper into mathematical understanding, and it gives students a visual aid that they can interact with. As long as the teacher knows how to use the technology and they have built their lesson plan around the technology, using it to the best of its capabilities, technology should always benefit the students' learning.

Week 5 - Math for All

Ever since I have decided to become a math teacher and started telling people that is what I am in school for I have received two different responses. I've either been told that I must be smart to be able to get a math degree or I have been told that that person never was a math person. Both of these responses hold math at a high standard that you have to be smart in order to understand math and that it is only for certain people who have the right genetics to be a math person. No matter how I have tried to explain it to these people that anyone can learn math, they don't believe it because they have grown up believing that math is only for some people, not for all of us. During this week's lesson it was particularly refreshing to hear that math can be for whoever wants to learn it, because our brains are constantly working and creating new synapses and connections and that all it takes is working at it to get the 'math brain'. It hurts me when I hear students say that they just aren't good at math because that means that they have given up on ever being good at it when all it is is that they are not good at it yet. Math is a process of learning, making mistakes and finding new ways to understand what you thought you already understood. We watched a video of Jo Boaler speaking about this exact thing; anyone can learn math and in extension anything that they want to, as long as they work at it. Now the key question is: how do we get students to not give up and be motivated enough to want to continue learning math?

Jo Boaler came up with a list of positive norms to help support math learning in the classroom, listed below:

These positive sentences can help to give a safe learning environment to our students so that they will be more open and willing to explore what they are learning. When I was in high school I was terrified of making a mistake or asking a question that all the other students might already have the answer for in front of the classroom. I was worried that I wouldn't appear smart anymore and because of it, I had probably lessened my learning. Appearance counts in high school; each student is worried about what the other students will think of them. If we are able to make social appearances separate from learning by making the classroom a safe place for the students where they won't feel judged by their peers, we can help them expand their learning by communicating, asking questions, and making sense of the math concepts. I believe these positive norms can help us to transform the math class into a true learning environment for all of our students.

Week 4 - Differentiated instruction

This week we talked about how to incorporate differentiated instruction in the math class. Before this year I wasn't sure how to fit differentiated instruction in the math class. Would I have to make separate lesson plans for the different learning styles or have dedicated time in each lesson plan towards each of the learning styles? I didn't realize how simple it could be to incorporate it into a lesson, such as the activity below.

With this activity we matched the different visual representations of each pattern. This represents differentiated instruction because there are many different methods to get to this final stage. Some of the students in class started matching graphs to the table of values because those were the two that they understood better. Others started with the equation and the graph because those made more sense to them. With each of these 4 representations, all students would be able to complete this exercise using their own strengths in mathematics. A lot of work went into creating this activity, but the work is definitely worth it because it helps each student to be able to perform this task.

The Knowing and responding to learners in Mathematics (2015) article talks about why differentiated instruction is important in a math class. High school students are growing at different speeds. Some may be farther ahead in their math understanding than others, so some students will be better at spatially understanding the mathematics whereas other students will be better at taking a visual representation of a math concept and connecting it to the abstract mathematics. By incorporating all of the different learning styles, we would be able to help each student to understand the math concepts in the lesson better. In order to successfully incorporate differentiated instruction we need to "balanc[e] understanding of mathematical concepts with procedural fluency" (1). Using the example above, we couldn't just get the students that prefer working with graphs to work only with graphs, they would still have to work with the pattern, table, and rule. However, by letting them choose which one to work with first, they are able to pick the one they understand best to help them work with the other representations that they don't understand as well, increasing their knowledge of the entire concept of linear equations.

I would like to do something similar to this in my future math class because I feel that it would help all the students to get a better feel and understanding of the concept.

Week 3 - Math Manipulatives

In class this week we worked with multiple manipulatives that can be used to explore different concepts in mathematics. When I was in high school we did not use math manipulatives often, even though it is in the curriculum. In my opinion that is because it is often assumed that the students in academic courses are more capable of understanding the concepts without the need of seeing it visually represented in front of them. I believe that manipulatives should be incorporated in every math course, for all levels of learning. This way the concepts are able to make sense in a concrete sense, as well as in an abstract sense.

When I have a math class I will try to use manipulatives whenever it would work in the lessons because they help us to have a better understanding of the math behind what is being learned. In class this week I was able to understand better the math behind the equation n*(n+1)/2 as the equation for the sum of 1 + 2 + 3 + … + n. Before this class I knew that the equation and the sum meant the same thing, but I did not fully understand why until I used the snap cubes to find out that by doubling the amount of blocks I had made a rectangle with dimensions n x (n+1), which is how we arrived at the previous equation. The snap cubes helped me to see the abstract equation in a concrete way so that 

I was able to better understand why the equation is equal to the sum.

Manipulatives can help any math student to have a better understanding of what they are learning and that is why I believe they should be used as often as possible.

Week 2 - Problem Solving

In class this week we were asked: 'would you rather have $500 or quarters stacked up as tall as you?' and later we were also given a tug of war problem shown below.

We were asked these questions and then left on our own to find a way to come up with a solution. With this method of teaching we were all able to explore our own way of problem solving. After each coming to a solution we then got into groups and discussed how we each came to our solutions. We found that there were many different ways to solve each of these questions all coming to the same answer. This shows that there is no one way to solve a math problem and that students should be encouraged to express their own ideas on how to solve the given problem

In the Dan Meyer video on the math classroom, he talked about taking the steps out of math problems and letting the students come up with the necessary steps to reach the solution. This method of teaching brings problem solving into the classroom and it also helps the students to get a better understanding of math. Instead of giving the students a system or formula to solve the problem, they have to figure out what information they will need to find the answer and what steps they would have to go through to get there. This promotes critical thinking and it helps the students to determine what knowledge is important and what is not necessary for solving the question. I would like to incorporate this method to student problem solving in my future math classes.

Week 1

Hello, my name is Colleen and I'm a teacher candidate at Brock University. I am in the concurrent education program year 5 and my teachables are math and French for grades 7-12. I have recently gotten married and I am excited to start this year off right. This blog is for the math education course EDBE 8F83 and I will be using it to keep track of what is going on throughout the coursework and to reflect on what we have been taught and how it could be incorporated into the classroom setting. I will be blogging weekly to see how everything will progress and hopefully it will help to prepare me for my placements later on in the year.

Throughout this course I hope to learn how to get my students motivated about learning mathematics. I have seen a stigma around mathematics that has made students disinterested in mathematics to the point of hating even the word "math". I would like to show the students that math can be fun as well as how math relates to the real world, while still teaching the students the curriculum that they are supposed to learn. I would also like to see the connections between this course and my previous math courses as well as my previous education courses to see how math and education can be combined to form the math classroom. I am looking forward to this upcoming year and all that it has to offer to prepare myself and the other teacher candidates for the following years to come.