Livin' in a Mathematical World

This week, I wrote my forum posts about Math in Life. I looked at Fibonacci's sequence of numbers and explored the way patterns can influence math. I really liked looking at the practice way patterns in math can help students learn.
Patterns in math make it more relatable. Math is not about basic rules, and more so about patterns. As an educator, I like the idea of thinking less about math rules and more about math patterns. By seeing math patterns in everyday life, students will be encouraged to learn and understand math in a real-life sense.
However, through discussion with my peers, I became interested in the math in technology videos. I decided to watch them again, and really investigate the way technology can influence math. Throughout our education, the importance of using apps in class is one of the most influential extra support to our students and their learning.
One of the things I noticed from the video is the use of outdated apps or websites that do not seem very appealing. I would prefer to introduce my students to more up-to-date apps because I think they will like them much better. Websites such as Google Play or the iTunes store have a variety of fun math applications that parents can download for students. This way, students are getting more reinforcement from home if they need it. A little extra practice may be necessary for some students to excel at math, and using technology based math apps will make kids think they are still having fun while they are learning. Check out this article on the popular education website Edutopia, which wrote an article about math technology that can be used to increase student understanding at home.
We also did a lot of unit planning. I have really enjoyed planning a unit on measurement. It has been very informative and I feel like I have learned a lot from it. We did grade 5 measurement and focused on the estimation expectations. These expectations are interesting for students because sometimes they do struggle with estimating a realistic number. Through our lessons, we tried to use a variety of means to teach students how to accurately estimate. I would love to implement this unit plan in a classroom as I feel it is a very strong unit. In the action portion of my individual lesson plan, I have students doing Minute to Win It Games and trying to estimate how long it will take them to complete the tasks. I think this is a fun way to reinforce realistic estimations to students, especially when dealing with time to the nearest second.
Overall, I feel that I have come a long way this year in math. I am happy that I have learned so many new and exciting things. I cannot wait to put them into practice in my upcoming teaching block!

Math Magical!

This week, I went to a Professional Development workshop on Math-magical ideas. The professor was so informative and really helpful. It was an awesome experience to be able to learn from her.
One of the things she did that really opened up my mind was start her introduction by explaining that she was not what people often call a "mathematician". She loved teaching literature. Her first love was books and stories. From this, she developed a way to teach math using stories and emphasized the correlation between math and literature.
As an English major in my Undergrad, I loved this! I was so excited about the book list she gave and all the ideas she inspired in me.

Retrieved From Google. (October 2017).

One of the books the professor was introduced called "Ten Times Better" (pictured above). Books like these use numbers inside of the story. Using this with any level can really help students have relational understanding. It can also be modified or edited to fit in every grade level. Perhaps before reading the below section, students could estimate how many teeth the dragon-animal has, and then what is ten times better than this? The cool thing is it is easily accessible for students, because most students enjoy being read to. Having a picture book allows for students to have a visual representation of the math problem that is fun and interesting, even older students will buy into this, because they will feel it is fun to be read to. Using literature to tell math-based stories is an easily modifiable task that can be rich and fulfill many expectations in one lesson.

Retrieved From Google. (October 2017).

Further, this week I was reading about demonstrating understanding through drawing in math. If students can draw out a problem and how they understand it, they are more likely to be able to get to the answer in an efficient and understanding way for them personally. I love the idea of using drawing in math to show interpretation and thought processes. It is an easy way to differentiate for students.
Drawing out math problems such as fractions allows students to have a real life example as well, because they can relate to it. Oftentimes, teachers use food examples when it comes to fractions because it is super interesting, gets students engaged and excited, and is an accurate representation of a fraction.
Using websites such as Math Playground to get students into math is also a good idea. Fun websites that are interactive and use play-based activities get students excited about learning. Sometimes they do not even realize that they are actually learning math when they are playing the games.
Another website we explored was https://nrich.maths.org which was a really good website for resources for all ages. I am loving compiling multiple resources an adding them to my library for reference later on. I really think they will be super helpful when in placement in a few weeks.
This week, I feel confident I have furthered my math knowledge and I am interested and excited to keep learning more!

Connections...

This week in mathematics we learned about making connections! I love using the strategy of making connections in language, so I was particularly excited about this video. I also watched TWO webinars on Differentiated Learning.
First, I watched Bernadette and James present on strategies and techniques for learning with differentiation. They provided awesome resources and I really learned a lot. Bernadette and James did a little trick on us and asked us a really complicated question that they would not repeat the instructions. It was so hard to try and answer this really long and complicated question when it was just asked out loud to us. I was so shocked and confused! But, Bernadette and James were just tricking us a little bit, because they really just wanted us to learn from that. They wanted us to think about the ways we could have changed the question in order to make it more accessible. They went through the three ways to alter a question: process, content, and product. How could the wording make the question easier to understand? How could it be presented in a different way? How could answering it be made easier? These questions got us all thinking. It was awesome to see all the suggestions we all had, because they would have really worked to make it easier for us to answer! In turn, that means it would make it more accessible for students to answer!
Secondly, I watched Kursten and Brittany. They talked a lot about ways to make an environment filled with differentiation. They gave a lot of practical examples and real life experiences. I loved watching their webinar. They talked us through different practical strategies that implement differentiation. My favourite was the Cube that they showed us, pictured below 

(Accessed through:http://www.guided-math-adventures.com/wp-content/uploads/Paper-Fraction-Cubes.jpg) 

These cubes all for CHOICE and flexible learning groups. I really think they are an excellent idea for math differentiation.

From these two videos the key thing I took away was the expectations of differentiation which means are:
Key Features of Differentiation:
1) flexible learning groups

2) choice

3) respectful tasks

4) shared responsibility


      This link takes readers to a Prezi I made last year on Differentiation. From this presentation, and now these webinars, I learned a lot about what differentiation means in a math classroom. In a math classroom, there are certain expectations that must be met. However, having clear ideas on differentiation make that a bit easier. When differentiating, its important to remember that Universal Design for Learning is better for everyone. It makes learning easier for all students! 

      Also this week, I posted in the 18X5 forum, because not many of my classmates chose to participate in the same forum as me. The 18X5 forum was about different ways to solve math problems. It reminded me of the thinking that happens when performing a math string. I loved reading this forum because it reminded me that sharing thought processes can really improve mathematical skills. 

   We also have an extra math seminar this week! It is a PD day where we learn about Mathe-magical ideas! I can't wait to learn more! 

Math / Mistakes &Success

      This week, we looked at making mistakes in math! Mathematics is one of the subjects that people often find it very difficult to allow for freedom in. Everyone believes that math has one right answer- and that is not necessarily true. Math work has multiple ways people can express the same numbers in different sequences or formats, and oftentimes the answer can be found in a variety of ways. It is important to recognize that creativity is available in math work, and if honed in on, can encourage students to make mistakes and thrive in an environment of imperfection.
       I also took the time to read the Math and Speed Forum because I feel like this is something I personally struggle with in math and was reminded of this week. During class, we did a Minds On activity that had us doing very quick math work on the fly, without calculators! This was something I had not done in a very long time. I was shocked at how long it was taking me to do more simple math without a calculator.
      Upon reading Charlene's post on the Forums on the Math and Speed section, I was enlightened by what she was saying. She stated, "speed does not equal intelligence" (Charlene Day). This is a super important thing to teach students when learning fundamental math skills. When teaching younger grades basic math such as multiplication and division, its important to repetitively practice at the students own pace to ensure that they have the skills before pushing them to do it faster.

This week was also the first week of Webinars! This took up a lot of my math-brain this week, because Charlene and I had to prepare our 30 minute interactive lesson. We had quite a challenge on our hands, but we pushed through and managed to come up with a Webinar based on creating a safe environment for math inquiry.
This is a topic I am pretty passionate about as a future educator, because I feel allowing for inquiry is super important. Inquiry allows students to freely explore what they find the most interesting in a subject. For mathematics, allowing for this to happen in an emotionally safe environment is extremely essential. Math somehow, and for some reason, often brings out the sensitive side in people, maybe because there is that fear of judgment in relation to a persons intelligence. This means that creating a positive learning environment for students is absolutely necessary to encourage a safe learning environment that allows for students to make mistakes! Charlene and I detailed a variety of ways to allow for safe exploration of inquiry. I am very proud of the work we did, and loved using the tools Charlene had explored during her first teaching block. I really learned a lot from the Webinar process, and loved facilitating it!
I also had the task of watching Ally and Nicole's webinar, focused on problem solving. I super enjoyed the way they played out their webinar! I loved learning about problem solving in a very realistic way. I really feel like I learned a lot of new math strategies this week.

Math-ittudes

This week was another challenging one in the world of math! During our online tutorial, I watched a really inspiring video about math mindsets and maintaining a positive attitude. This is something that I think is super important as a future educator. I want to be the kind of teacher that encourages that little voice in my students head that says, "I can do it if I work hard!" I want my students to maintain that growth mindset, and feel like they can take on the challenging work of mathematics.

I also read an article about spatial reasoning. I chose this article because I do not have a lot of information on the topic, and wanted to learn more. The document states, "Spatial thinking, or reasoning, involves the location and movement of objects and ourselves, either mentally or physically, in space. It is not a single ability or process but actually refers to a considerable number of concepts, tools and processes" (3).  The benefits of studying spatial reasoning are clear in all aspects of mathematics, as the concepts often intertwine with one another and all of them work together. As stated in the article, "By exploring the spatial aspects of mathematics, we make it more accessible, more engaging and more relevant" (4). Below is a picture I took from the article that I found very helpful. 

Support Document for Paying Attention to Mathematics Education (4). 

Further, I found that this week, I am really starting to notice the way that math works together to assist students everyday life. Creating connections to real life allows students to find meaning in mathematics in an easier way. Also, using creative ways to present information will allow students to subconsciously learn math, like through the card games we have been doing as a Minds On portion of our classes. Metacognitive thinking allowed me to acknowledge the way I am learning in math lecture, and apply it to how I want to teach in the future. I plan on being the kind of teacher that uses positive words, that acknowledges the negative but tries their hardest to make it into a positive.

Back @ it !

Almost a year later, and here we are again, bloggin', teachin' and doing some learnin'.

I am excited at the opportunity to really work on my blog this year and make it look pretty, as well as flush out some math content on it! I am hoping to employ Charlene to help me! (Fingers crossed!) 

This week , we began our math class again, and I'm not going to lie, the first class was a little stressful! We had a lot of stuff to organize, groups to get together, and math to learn. I am once again scared at the thought of being in a math class again, but excited and feel more prepared this time around. 

At the start of the week, we were assigned an online module in place of our Monday class. This is excellent for me, as a commuter and person who loves online learning! We got to watch a variety of math videos, all of which I loved and each which had its own excellent and thought-provoking message. My personal favourite is this one: 

     I personally like this one because while I was watching it, I was thinking of all the ways students eyes could be opened up to the way they see math, and why they see it that way. I was also thinking of the cross curricular intersections I could do with math and language based on a video like this! How awesome would it be to be able to write about math perceptions from a social media/ TV and movie perspective. I think the students would have some very interesting opinions on a video like this.

Attitudes Towards Math. 2017. Retrieved from Pinterest. 

         In class this week, we really discussed dispelling myths about math. I loved having an in-depth conversation with my fellow teacher candidates about why a large majority of us feel uncomfortable with the idea of teaching math, especially to higher grades! Really getting into why we still hold onto these math myths that we have been taught at a young age was enlightening.

       As we move into the semester full force, and I prepare to teach math in my second teaching block, I am really hoping to remember not to fall into these math stereotypes. There is no such thing as a "math person". And one of the biggest things I learned this week was that there is still a very deep need to dispel gender stereotypes in math. The fact that students as young as grade 2 are identifying and associating "boy" names with math, and "girl" names with reading is astonishing. As a teacher and a feminist, that is something I would really want to work to break down. I would never want one of my students thinking, " I cannot do this because of my gender". I really look forward to the day when I can make lessons that work towards creating students with a positive math mindset.

reflecting

Course Wrap Up! For Mathematics 

This course was a very beneficial experience. I have enjoyed learning how to teach mathematics. I came into the course very afraid of what would be expected, because it had been so long since I did math. I have found a lot of very beneficial games and activities from this course.

I also found the textbook to be incredibly useful. The textbook often detailed misconceptions, activities, and very useful observations that previous educators had witnessed. I loved the textbook because it covered all the bases. It was also extremely useful in showing us how to teach mathematics. 

My views on mathematics have changed so drastically. I am still very afraid to teach, but now I feel more prepared. I did not have the skills or knowledge I did before taking this course, and am very happy to have had this experience. 

This week in class we played a very fun math game that can be applied to any kind of holiday or special event. We really loved it as a class and I cannot wait to try it out. The game was a strange version of Battleship. The class was supposed to place the little boxes of options onto a grid and then as the grid squares were called out, the students X out the boxes and either gain points or rewards or lose the numbers they have gathered. I thought this was super fun and hope to get a chance to use this in my placement.

This week in placement I got to see another open question where students tried to answer questions about how two cylinders have the same surface area and different dimensions. From this question, I noticed a lot of students struggling to understand the difference between 2D shapes and 3D shapes. The students were attempting to calculate the 3D shape of the cylinder without realizing the surface area meant it wrapped all the way around. They tried to calculate it as two circles and one rectangle. The teacher eventually brought out a soup can to demonstrate. It was awesome to see some students had already created their own manipulative using paper. This helped them a lot. 

I am really looking forward to exploring math more and am thankful for the knowledge this course has taught me. I know that I have much more learning to do, but now I am excited instead of scared. 

there is 100% chance I am writing a blog right now ... (week 10)

     This week is all about probability! I actually like the topic of probability because I always remember doing fun activities such as rolling dice, flipping coins or playing carnival games. But using the actual mathematics behind probability is a bit more complex than just games!
     One of the things I feel probability really strongly assists in is demonstrating the relationships between numbers. When Michael did his presentation, he explained that most students can have some basic misconceptions about probability variations. If you flip a coin 500 times, you still have a 50/50 chance of it being heads or tails. There is not other option, and regardless of how many times that coin is flipped, the odds remain the same.
    Another aspect we discussed was taking this information and graphing it or placing it into charts. In my placement class, they are always using T charts or other visuals to show their work. The student seem to really thrive when they are given a clear display of the numeric relationships in chart format so I am constantly looking for new examples. I will definitely be borrowing these ideas when it comes time for my placement.

"Trapezoid" Varsity Tutors. 2016. Retrieved from http://www.varsitytutors.com/advanced_geometry-help/how-to-find-the-area-of-a-trapezoid

    In placement, the students are still working on area and conversion. They really like seeing relationships between numbers, and we have continued through math strings. One of my favourite lessons was based around an open problem with the grade 7's. I had the opportunity to split the class up with my associate teacher, and he taught the grade 8's a lesson, while I supervised the grade 7 math problem. He called it an "open problem" and it was basically a generalized question that there were no real, cookie-cutter answers to. I cannot recall the full question, but know the students had to attempt to find a formula that would calculate area of a trapezoid. There was not any more information given. From this, they split into pairs and tried to come to a solution. It was great to see the students work through the problem on their own. They did want help and guidance, but not too much. They knew to channel their frustrations, which was so great to see. I also liked the concept of giving students freedom to explore in mathematics. I feel that students need to understand and make up their own relationships when it comes to numbers. To see these students work through a problem with very little guidance, and come together to learn off of each other, was the kind of math problems I hope to implement in my classroom one day.
     I found this resource from the same author as our textbook, I would love to read more about open questions and see the impact they have on our students.

measurement (it rules!)

This week started with me preparing for my presentation on measurement! We had no in class session this week, so I had to adapt my presentation to fit a smaller group. It was interesting to see how being adaptable is still one of the skills I am learning. I naturally like to plan ahead, so when my plans change, I sometimes don't know how to accommodate those changes. I am still learning!

         In my presentation I had students draw out their hands on 1 and 2 square centimetre paper, and count how many centimetres their hand was on each paper. From this, I hoped they would gain an understanding of measuring in square centimetres, what square centimetres are, and how to compare two measurements.
         It went pretty well though. It is also very interesting video taping yourself and being able to look back on what you said and did. I think I would edit some things if I did present this lesson again, but I am happy with the activity,  it is always interesting to get students to understand how they can use the world around them mathematically.
         One of the things we learned online this week was investigation is one of the key aspects of measurement. Our professor discussed how students may not have the basic skills of measurement, and I think that is where my presentation would come in! It is a starting point for measurement.
The topic of measurement has many different features, because one can measure a variety of things. This features include measuring temperature, time, area, perimeter, circumference, and volume. One of the resources I looked through while planning my presentation was the Measurements grade 4-6: a Guide to Effective Instruction in Mathematics, Kindergarten to 6. This guide features really good examples for educators. It is laid out with similar sections as a lesson plan, making it super accessible and easy to read. One section that stood out to me is that it provides specific examples on how to scaffold the problems. There are several examples of "scaffolding suggestion" boxes featured in the document that seem really useful. On Page 49, the document suggests ways to connect a linear representation of time to life events that students will understand, in the scaffolding suggestion box. I think these little ideas and suggestions to educators makes it easier for pre-service teachers to understand, and have ideas to reference. I love documents like this because they take away the fear of trying to implement complex topics scaffolded in effective ways.
         I also sat in on Kursten's presentation, and assisted her in filming it! It was a great experience to see her activity. She also did hers on area, but focused on estimating area. Her activity is also a good introduction or refresher for students. In her presentation, the students cut out pictures and estimated their size based on the other pictures they cut out. This was an excellent example of estimating area! I definitely saw how hard gaining a conceptual understand of measurement can be for students.

snakes on a plane (week 8)

This week was about geometry and spatial sense. I actually do not mind this unit. I like thinking of how things can work together spatially. While reading through the "guide to effective instruction: geometry and spatial sense grades 4-6" teaching guide, I found a lot of information that I could see being very useful when it comes time to implement a lesson. One of the concepts that stood out to me was the inclusion of teaching strategies that are effective for this particular math lesson. They included several examples of instructions on what methods to use to teach geometry and spatial sense such as, "carefully planned activities will enable students to build on these [personal] connections and identify relationships between and among the various areas of geometry and spatial sense" (25). The document contained ample information on how to teach, the basis of why students need to know this information, where the information came from, and most important of all, how to connect it back to our students in a way they will remember. I will definitely be using this guide in my practicum. 

"Geometry Pun" (2016). Retrieved from tumblr.com

Geometry and spatial sense are so essential because they are skills that can translate to everyday. We use shapes in our everyday life, and need a sense of where things fit in the world. There are many real life examples and connections to be made on a unit like this. 
One of the things discussed was the misconceptions regarding comparing different units. For example, students need to be made aware to be careful of comparing shapes with different units. In my presentation, I introduced students to measuring perimeter with 2 inch grid paper, and 1 inch grid paper, an example like this would help students to see the difference measuring with different square centimetres can make. (featured in next weeks blog)

I will work to make sure my students have a conceptual understanding of geometry and spatial sense by providing them with a variety of demonstrated examples, and scaffolding the work to make it easier for them to understand. With a unit like this, which has foundations all the way in ELKP, if a student has missed a concept or doesn't quite understand something, they will struggle to build up any understanding. I will also focus on the big ideas. As was detailed in the effective instruction document, "These big ideas are conceptually related and interdependent, and instructional experiences will often re ect more than one big idea. For example, when students create or analyse designs made by transforming a shape or shapes (location and movement), they demonstrate an understanding of congruence (geometric relationships), and of how congruence is connected to the properties of a shape (properties of two-dimensional shapes and three- dimensional cues)."  (16). This quote stood out to me because it demonstrates the way every step of a lesson can affect a students conceptual understanding of the whole. 

I enjoyed the presentations this week because I loved playing with the shapes and seeing how that worked. I particularly enjoyed Ashley's presentation because I felt it was a good lesson for junior learners. I liked the use of manipulatives. Our table actually thought making the shapes and putting them into the bigger diamond was going to be very easy, but it was actually harder as she gave us parameters to follow. 

I also enjoyed learning from James. I liked his trick with flipping the paper to be able to see how shapes are translated and flipped and can move. I tried out the method of doing the exact opposite coordinates, and am happy to have learned that because I feel like I can really use that in my practicum. I do sometimes get lost when it comes to translating shapes and moving them across the grid. While working with students, I often try and get them to cut out a paper or have a physical resource so we can actually work with it, like Jacob did. Overall, I really enjoyed the presentations this week. I have been learning a lot from them. 

This week I am working on my own math presentation on measurement so stay tuned for updates on that! 

let's taco 'bout it ! (and by it I mean patterning and algebra) (week 7)

     

"Let's Taco Bout It" (2016). Pinterest via Hollister Co. Retrieved from https://www.pinterest.com/pin/56154326576907607/

       This week surrounded patterning and algebra. When it comes to algebra, I am automatically lost! I feel like I lack the foundational knowledge to build up my current skills, so I am trying to re-learn and teach myself a lot of these concepts. It has been a challenge.
          We started our learning with a Speed Dating Multiplication number sequence. This activity required us to each start with a number, then pair up repeatedly to have two numbers to multiply together each time. I enjoyed this activity, while I did find it challenging. We were multiplying pretty high numbers, so it our answers were in the thousands. I did like this activity as a "minds on" portion. It got us thinking about multiplication and the relationships between numbers. It is a good example because it can be applied to any grade level. Even grade 3-4's could do this worksheet, the teacher would just give the students smaller numbers to work with.
        While discussing patterning and algebra, one of the things I found was a lot of great activities. The textbook provides a variety of image examples to show patterning. The misconceptions students often had was related to the changes in orientation of a shape. I think this is extremely common, as students often think the way a shape is oriented defines part of it, but this is not the case. Two shapes can be exactly the same but mirrored, or translated. Using manipulative shapes would be a very good way to avoid this.

          One of the things that stood out to me this week was Mohamed's presentation. He was very funny and engaging and I liked the way he presented the equations. I felt like I understood and followed his presentation. One of the main things I was happy he went over was the difference between an equation and an expression. As a student, I struggled with this concept. I often could not identify what the question was asking, which would leave me lost. I liked his distinctions because I think comparing them made them stand out better in my mind. Students would take a lot out of his presentation. I also loved the GOAT! Greatest Of All Time would be a really funny title to introduce to students and let them compete for the title. I also loved that he took the time to relate his examples to real life experiences. I always remember things better when I form the memory around something I already know and understand, so Mohamed's presentation stood out to me after he made the concepts relatable to us as students and educators.
        I also enjoyed Nicole's presentation. She related the Fibonacci equation to the real life Parthenon. This task was challenging and hard to understand, but created a great learning environment. She explained the equation in a very understandable way, and I think if I am teaching that concept I will think back and use some of her ideas. Real life examples stick out in any students memory much more than memorizing an equation will.

"The Parthenon and Phi: The Golden Rule" (2016). Retrieved from Google Images. http://www.goldennumber.net/parthenon-phi-golden-ratio/

      One of the other concepts we discussed was teaching students who are learning or have learned English as a second language. This was very useful and can be applied to any subject area. I love when we discuss problems like this because it feels like very relatable information that I can see myself using out in the classroom. We discussed the challenges that come with teaching ESL students, but also were provided with strategies to consider. I have worked with a few students while volunteering that are ESL learners. I found that cues and gestures worked best for me, as I am already one of those people who talks with their hands. One of the things that stood out to me most of all was the idea of a language directory/ personal language dictionary. I loved loved loved this idea! I cannot wait to try it out. I may even apply it to an English class and make them all do it, regardless of their first language. The idea of students keeping a book that they write words they do not understand, may have heard in a different context, do not really know what it means, feel like they need to engage with it more is so exciting. I think it would be really beneficial to do.

"Lesson Plan ClipArt" 

      So far, I have found the hardest part of planning a lesson to be getting started. I feel like I do not know where to start. I have seen these great ideas, and understand the curriculum documents, and have started to become familiar with the process as a whole, but I still feel unsure about where to begin. It is a very intimidating process, because it comes with a lot of responsibility, and I want to do right by my students. The terminology in lesson plans has also been challenging to distinguish. We have gone over it once in our practicum class, but I still do not feel like I have a clear understanding of the terminology. I am also struggling through picking out curriculum specific and overall expectations, but am learning that and feel like I have clear expectations on that part.
        To make lessons learner-centered I plan on considering my students first when planning a lesson. I will make sure to be critical of ideas and think about my students when deciding whether they will work or not. At its core, lesson planning should first and foremost consider the effectiveness of teaching to these particular students. One cannot just take a generalized lesson plan and apply it to all students because not all students learn the same way. I hope to take each student into consideration when planning lessons.
   

6:12 (ratios and proportions) (week 6)

     This week we discussed ratios and proportion. To ensure my students develop a full understanding of these concepts, I plan to use the strategies we discussed in class. One of the things we did was use real life example, such as baking to understand how we need to proportion things out. Guliana used this example to demonstrate how chefs and bakers need to know their proportions and ratios because they are constantly asked to very quickly alter recipes in order to feed less or more people. This means having a complete understanding of the numbers and how they are balanced in order to make sure there is just enough ingredients. An example of that is if one adds to much salt to a batch of cookies they will not taste good. If we are doubling the batch, we have to know how much to add to each individual ingredient so it does not negatively affect the whole. The same works for ratios. Ratios are a good base for comparing things. They allow students to have an understanding of how two numbers can be contrasted against each other, or compared. This will allow the students to have a better understanding of how numbers work. The relationship of 4 blue squares :2 green squares can also be thought of as 2 green squares: 4 blue squares. This could also be expressed as a fraction, 2/6 or a percentage, 33.33%. 

6 Squares. (October 2016). Casey Made with Pages. 

   

    We also looked at percentages, because the topics are so closely related and interchangeable.With percentages, I always find the visuals a very good representation of determining and understanding percentages. I find this unit to be very important because it is extremely useful in everyday life. Even to this day, students are given marks out of 100%. This means that if they have an understanding of what a percentages is, they can take the mark of 25/40 and know that they got 62.5% on their test. Practical skills like that are always useful and can be applied in various situations. In the textbook, it does state that one of the common misconceptions is students who do not have an understanding of percents that are above 100%. Explaining this as equivalent to a mixed fraction would be a very good strategy. I think I would also try and draw a visual, like one of the grid visuals used throughout the textbook, to demonstrate what it means to have more than enough, and to show how that works as a mixed fraction. With these, pie graphs are also a great visual, for another class I made a pie graph about students who wanted to join a book club: 

    This was a silly joke, because I obviously did not actually ask 90 people to join a book club. But the percentage needed to be out of 100%, and I wanted it to be obvious that it was many more people that did not want to join, so I took the numbers of 60 people who did not want to join, 3 people who wanted to hang out, and 2 people who wanted to join and multiplied them all by 2 so that my ratio could be out of 100%. The initial ratio would have been 60: 3: 2. This pie graph is a good visual because it provides students with a representation of parts out of a whole. If they can see that the numbers represent something, they will have an easier time developing a conceptual understanding of these concepts. 
    In Chapter 16 of the textbook, a section was dedicated to Appropriate Manipulatives. From this, I found my understanding of teaching this subject was greatly increased. It listed counters, cubes, tiles, or balancing scales and paper bags. I think the activity with balancing scales would be really fun for students to do. It would provide a real life example they can understand. The algebra tiles would also be a very good manipulative for this topic. They can be used to show how the formulaic expression is equal. I think using these manipulatives to show ratio and proportion would be great. 

    To ensure my students misconceptions are addressed correctly I will listen closely as they explain things. I plan to develop resources in the classroom that ensure that each student who is misunderstanding something, or feels they do not have the right tool to address the problem, will receive extra guidance and further explanations. Being able to understand their misconceptions is the first place I would start, and as someone who will have had the same misconceptions as a math struggler, I think I will be able to see the thought process behind some of these students. I always found I applied the wrong concepts to math units. I will attempt to learn from the student and see how they thought out the problem so that I can then address it. Group work is another great resource so that the students can then help each other out. Hands on projects and real life examples are my favourite ideas to developing students complete understanding of concepts. Ensuring the students understand will come from practice and continuous learning. 

positives and negatives (5)

       This week was all about integers! We talked extensively about how hard it can be for everyone (students, teachers) to understand the concept of negative and positive numbers. They can get lost if we just show an equation with a mixture of integers and don't have a background explanation for what is going on.

     It is always hard to find real life examples that make sense to students but also teach them something they will remember. For this subject, I loved all the cool ways we learned how to keep track of the positive, negative relationships and their outcomes. One of my favourite graphics is this one: 

"Integers"" Don Steward. (2016). Retrieved From http://donsteward.blogspot.ca  

    Finding unique yet relatable ways to describe and explain things will be the most effective way to reach the students. I find that real life examples with a combination of pictures and words help me, so I will try and access my students through those means. I want to appeal to all learners while providing them with challenging material. I think the above image is a nice place to start, with that image I would then include temperature and think of all the ways I could demonstrate how integers actually visually work. One of the other ideas that was suggested is a sideways number line. I personally love number lines and how they provide a nice clear visual of what the numbers are doing, how they are moving or jumping around, so I think that could be very useful.
   One of the things discussed in the textbook was around the misconception of "two positives makes a negative" (351). Sometimes boxing students into rules makes them forget the other aspects of the question. In this situation, the students can forget that the rule applies for multiplication and division instead of addition and subtraction. This is an example of students not thinking about the real world applications and what the question actually means. 

"Printable Number Line- Vertical" (2016). Retrieved from https://www.printablepaper.net/preview/Numberline-Vertical

    Another one of the activities I really enjoyed was a game that was a cross between Jeopardy and Battleship. We had math equations that we had to quickly figure out that featured both positive and negative numbers, and then had a sheet with the answers that acted as our Battleship board. It was a fun way to get students to really quickly do several math problems. I found it was very helpful at reminding me what adding and subtracting positives and negatives entails. I definitely would consider thinking of a similar game to play with my students.

     We also talked about exponents this week. I do not mind exponents, because it seems to be a very straightforward topic to me. It is a set of rules that are to be followed, and because of that we have to ensure that students remember what the exponent represents. 

"Exponents". Sixth Grade Math Blog. (2016). Retrieved From http://hemsmath6.weebly.com/exponents.html

    

    

       I think like most topics, this is one that just needs to be broken down into smaller steps for students to understand. It is easy to do once we have the base of remembering the rules and knowing the terminology. For a topic like this, I think it is really important to spend time on the base work such as fully understanding the terminology used to describe these numbers. From there, I would try and explain why we simplify by making it an exponent, like the above example demonstrates. If students first have the basis of when and why to use exponents, they will then know what to do and how to do it to solve the problem. 

   

    One of the other topics we discussed was manipulatives. I know how hard it can be to decide when to introduce manipulatives into a topic. When assisting grade three's in learning multiplication, I had chosen to introduce manipulatives far too early in the conversation, and found my students did not have a good enough base to work with the manipulatives on. It was a hard lesson to learn but I definitely noticed right away that they did not have that foundation, and then the manipulatives because useless instead of being a good tool to use. I think teaching the lesson with demonstrations and visuals is always a good thing, but the students need to have that clear outlined lesson first, where they are taught how to do it, before giving them the tools to explore how it works on their own. The manipulatives are an amazing tool for kinaesthetic and visual learners, but I think mathematic lessons need to be fully explained and detailed out to students before they get to that point. 

   Overall, another great week with various advances in mathematics. I am thrilled to be learning more ways to teach and hope to continue having fun and inventive examples to present to the students. 

My Math Forum Question

Due to the upcoming Thanksgiving vacation, I am planning a road trip to Kitchener- Waterloo, Ontario to visit friends and family. My starting point is Fort Erie, Ontario. 

Delicious lasagna that is awaiting my arrival

1. Please draw the route that I should take on this map 

B) Now that we have the route planned out, we need to figure out how long it is going to take me to get there. The trip is 165 kilometres one-way. I have to be there in time for dinner at 5:30pm. What time should I leave home at if I am driving 100km/hour constantly throughout the trip?

C) What if I decided to take the back roads, and would have to alter my speed and travel at 80 kilometres for the trip? What time would I have to leave home at then? 

D) The next step is to figure out how much money I will need for gas. I have a small car that holds 76 litres of gas in the tank at one time. I plan to fill the gas tank to full before I leave, gas is priced at $0.94 per litre. How much will it cost me to fill up my tank?

Fractions and Decimals! (week 4)

I will ensure that my students have a conceptual understanding of fractions and decimals by displaying to them the basic concept of both terms before elaborating on those concepts. I think that most confusion can stem from not having an understanding of why those numbers are in the place value they are, as well as what it means to switch fraction to a decimal and vice versa.

The most basic part of understanding fractions is realizing what the numbers mean and what they actually represent. I will work to show my students that through interactive conversations and examples. The textbook provided some great activities involving games and creativity that I would consider using. The pizza game we played in class, where we all started with a number and that number had a portion of the pizza such as 3/4ths of a pizza, and that had to be compared to everyone else’s number even though they had different denominators, showed us the importance of having the same denominator to compare fractions. I would definitely want to try to do something like that to start off my students, to make sure that they fully grasped the concept of fractions. 

As far as decimals, I find the place value charts to be the most effective for me so far, so I would first try that with my students. Placing the decimal into a chart and demonstrating the movement of the number while having the title of “tenth” or “ones” columns was very helpful to my understanding. I think that creating a place value chart would be the first place to start with students. I have done this with the grade three students I work with, and they use a thing called a “place value house” and act as though the columns are neighbours and when they have to borrow it is the neighbour knocking on the door asking to borrow some eggs. This was a good technique because it reminded the kids of the values in the columns. 

I also really enjoyed Nicolina's example of having to match the fraction to the decimal. It was very similar to this online game, that I found when I was looking up fractions and decimals.  The matching game of having a mixture of fractions and decimals and having to match the equivalent fraction to its decimal was a quick, fun way to practice conversion skills.

Fraction Pizza. Clipart Kids. (2016). Accessed through : http://www.clipartkid.com/pizza-fraction-cliparts/

In my placement, I have yet to see the students work on fractions. They were working on scatter plot graphs when I went in last time, and it was really interesting to see them work through the problems. I started with some questions that were actually kind of hard to answer, luckily after talking it through, my memory was jogged to remember how to create the algebraic expression and keep the y axis the same while changing the steepness of the slope. I cannot wait to find out some more techniques that I can use in the classroom. I have tried to implement a few so far, such as encouraging students to work through the problem in pairs. I have found that this math course is really helping me in my placement since they do math the first two periods I am there. 

This week in mathematics ... (week 3)

This week I had my first day of placement in the classroom. I am in a grade 7/8 classroom at a very nice local school. I am super excited to see what the next few weeks with this class will bring. That leads me to one of the things I want to talk about this week ... math strings!!
When the students got to class, the first thing they did was pull out a book and started doing math strings. The teacher would write out any random multiplication problem, and the students would raise their hands when they had the answer. The teacher was making them up on the spot, as he had done all the ones from the textbook they are working off of. For anyone who does not know, a math string is a tool used to encourage mental math. The problem is started with one multiplication question, but really it could be addition, subtraction, division, any type, but in this case they did multiplication. They took the first set of numbers, and found the answer. Then moved to another question, this question usually featured one of the same numerators, for example, if the first problem was 2X2 =4, the second question could be 2X4= 8, then so on and so forth. It does not have to be done so neatly, as you could just to a number like 2X14= 28, which the students could use the previous "string" to determine using mental math. The other cool thing is when the teacher called on the students, they are automatically expected to explain how they got their answer. So all of the mental math strategies are being shared right there, on the spot. They are also apparently called "number strings". This blog had a lot of interesting explanations on number strings for younger students (grades 4-6). I researched through it a little and found that they use a more structured approach to their number strings, but the teacher I was working with had done so many that he did not need to pre-plan out his strings, he just worked through the numbers as he went.

     Furthermore, I am interested and excited to see how the strategies I have been learning in math class will help me when it comes to my practicum. One of the concepts I enjoyed learning about was the "representation" concept. If students are comfortable seeing numbers as more than just numeric values, they can work with them in different ways. The example of 9 is the same as 10-1, or 8+1, or 3X3.
     One of my "aha" moments this week was a lesson we did on inverting fractions. The idea that we just invert and multiply had come back to me during a required pre-assessment course. I had remembered what to do, and when to do it, but the "why?" stuck with me. Unfortunately for me, as a math student, no one had ever taken the time to explain why, it was always just well here is what you do, so now go do it. Being explained why was really awesome. So, for the inversion of fractions, I now understand that we do it so that we can compare unalike fractions in the same way. This video works to explain it better than I can, but the gist of it is that to compare fractions, they have to be "out of" the same thing, or else the comparison is meaningless.
     One of the other things I was relieved to have learned about was that assessment for learning is not just a ranking of students, but rather a summation of their strengths and weaknesses, which are build around trying to help the students improve rather than to just judge them, slap a grade on a paper, and move on. With math, where simple yes or no right answers are available, this sometimes gets lost. I am thrilled to have the "good assessment plan" guide at my disposal now, because I think it will be really useful. It was also interesting to consider that when students do bad on an assessment, that teachers should reflect on that. Why did the student do poorly on the assignment/test? What could have been explained to them differently? What could I have done as an educator to ensure that these students succeed? I think these questions are all very important for us to ask ourselves every day, especially when our students are not understanding the concepts we are teaching.
   I also really like the idea of instant feedback in seeing if students truly understand what they have just been taught. The use of Twitter or a face-paced multiple choice test could be options. Technology would be a great form of instant feedback, that could help struggling students who do not feel comfortable speaking up in class.
    Overall, I feel I have made many strides this week, I really enjoyed reading Chapter 3 of "Making Math Meaningful to Canadian Students, K-8, third edition" in particular, because it discussed the pragmatic behind the sense work behind teaching such as the grading schemes and strategies in lesson planning. Each week, more of my math questions are being answered, and I am hopeful that in two years, I will be somewhat of an expert!
    Another exciting thing we did in math class was students doing presentations. One of the students did a really cool presentation on multiplication, which was really engaging. I liked the assignments where we broke down the concepts and used them in practical ways. One of the presentations used number place values and we had to take certain numbers and use them in their correct place value using different combinations. This ended up being a really fun assignment, as we all raced to see what combinations we could come up with. These are the kinds of assignments I hope to include when I am an educator, because we all learned about place values while having fun. I definitely enjoyed the presentations! I think it will be great to continue seeing how my peers use various tools to teach math, and I am hoping to borrow some of those when I am an educator.

Citation

 Small, M. (2013). Making Math Make Sense to Canadian Students K-8 (Third ed.). University of New Brunswick: Nelson Education.

Another week, another reflection (2)

          One of the resources I found out about this week was called a “Settler”  a short quiz that features multiple “simple” math questions to see where students are in the curriculum. For example, the settler I completed had a series of subtraction questions that led to a variety of answers that added up to give one total answer. We had a short amount of time, maybe two minutes, to complete the settler and move on. It seemed like a good way to start students off in thinking about subtraction, which we could have built on in the lesson and done division or something similar. It also contributed to getting our brains working in a mathematical way. The only thing that worries me about such an activity is the added stress when students feel they have to complete a large amount of work (from their perspective) in a small amount of time. I know when we used to do activities like this when I was young, it caused me to compare how fast I could do math to others, and when I couldn't keep up, instead of pushing through, I shut down completely and didn't even want to try. This was especially true because the quick math students were always praised and rewarded for finishing their math quizzes first. That now encourages me to seek different means of finding active learners who can persevere. I would maybe try and do these with an older group and see how they reacted first. I would also take away the blatant reward for the person who finishes first. Perhaps even get students who struggle to first work in pairs, then over time, ask them to slowly start to try the settlers on their own. I definitely think this could be a useful tool, and am still hopeful that the reaction to it afterwards is what discouraged me when I was young, so now I will try and find a way to effectively encourage and engage all students with an activity like this. Below is an example of a settler... 

        

Math Settler. (2013, February 10). "Settlers"[digital image]. Retrieved From http://www.scoop.it/t/mathematics-starters/p/3996398803/2013/02/10/settlers 

  I would like to continue to learn better post-lesson reactions. The main thing I am still trying to grasp is how to react to students who are bad at math without singling them out as “slower” or not able to keep up with the brightest students in the class. I think there has to be a way to alter the lessons in order to encourage learning in all students, regardless of where they start off or their basis of knowledge. With math, if students lack the basis of knowledge, for example how to add quickly, multiplication becomes extremely difficult, then if they continue to struggle with the basic multiplication, they will never be able to move on and do algebra, or statistics, or any other stream of math that they need that basis of knowledge for. The idea that students are being left behind because they are not speaking up when they do not understand something, or need more help, is something I would like to be able to spot out, and then help them learn the necessary skills to move forward. I think a good student is someone who says they need help when they do need it, but an excellent educator can spot that student out without being told. 

After reviewing the “Ontario Grade 1-8 Mathematics Curriculum” document, I was surprised to see the accountability the curriculum places on not only students but teachers, principals and parents. It was nice to see that they hold students accountable for their learning, but also encourage secondary learning sources such as parents and the principal responsible. If a variety of students are not keeping up with the mathematics curriculum, then perhaps there is a pedagogical reason. While this document contains an overwhelming amount of expectations, it seems to be an excellent guideline for teachers to follow, and I enjoyed becoming more familiar with it. 

     I also found some pretty cool resources while researching "settlers" a bit more, one of them from Statistics Canada. This would definitely be a good resource for Junior/ Intermediate mathematics. 

Reflection One (week 1)

Chapters One of Making Math Meaningful to Canadian Students K-8, by Marian Small

starts with an introduction that is very comforting, as it breaks problems down and explains how that helps students. The example of 3X5=15 from page 3 was a great way to put it. A student who understands that multiplication is asking for 3 groups of 5 will understand all multiplication, while a student who just memorizes won't necessarily develop a deeper understanding. 

             One of the other things that stood out was: "understanding is most likely to develop in classrooms that focus on problems to be solved, rather than exercises to be completed.” (Small, 3)  This is very interesting because it aligns with the kind of educator I would like to become. 

Another one of the first things the textbook talked about was math manipulatives. This is something that was stigmatized when I was younger, like counting on your fingers, or having to write out all the steps to a problem, which increased my confusion and prohibited me from obtaining deeper understanding. I am thankful research has improved and that math manipulatives are now accepted and encouraged. While volunteering I witnessed students using them with great success, specifically on the grade 3 EQAO math test. Overall, I am excited to learn more about new and innovative math manipulatives and to continue breaking down the stigma about needing that little bit of extra help with math problems. 

Math Manipulatives. (2006-2016). Rainbow Resource Center. Accessed through http://www.rainbowresource.com/product/sku/MMMSST/

The other section I enjoyed reading was “Different Ideas to Many Strands” (Small, 18). This section explained how to take a suggestion that a theorist many have about math, in general, and to apply it to all the different types of math. If a student can understand the broader idea as a whole, they will find it much easier to apply it to any different kind of math question. The example the textbook used was how certain classes of objects behave in similar ways, when they applied that thought to geometry and fractions they found a pattern in how it appears. This connection can be invaluable to students who are struggling to apply big ideas to small concepts. 

Another thing I was interested in reading was from the “TIPS4RM: Mathematics Processes” article in which it detailed some core basics involved with learning mathematics. The subtopics included things such as: reasoning and proving, problem solving, reflecting, representing and connecting. These are all considered mathematical process skills, which the article explains, can empower students to have a positive outlook on math. The idea that small changes in the way we teach students math can change their entire outlook on the subject is very exciting.  The article uses charts provide a really excellent guide that will come in handy when I try and infuse these “big” ideas into any math lesson. One of my favourite subtopics was “connecting” because that is what I struggled with most as a student. I think making the connection between the lesson and actually doing the problem on my own was the hardest part. I did not find that the teachers knew how to support me in finding links between the concepts being taught and those I was expected to enforce. This section helped me see how I could use different strategies to prevent my students from falling into the same pattern. The connections between math, real life scenarios, and other disciplines is so strong, and I think emphasizing that would be really beneficial to educators and students. 

               These strategies all seem to be working towards students having greater understanding of why we do the math that we do. Instead of abstract work and tedious word problems that do not relate to everyday life, these theories and tools have encouraged me to a more positive outlook on mathematics and how they are being taught in a more positive, connecting, reflective way to students. I am hoping to use some of these and test them out for myself. I can’t wait to see how students react to these concepts!