Mathematics Week 6

Math is EVERYWHERE... 

Math in Life 

Youtube Account (2013). Fibonacci Pine Cone. (Online Image).
Retrieved from:https://www.youtube.com/watch?v=nt2OlMAJj6o 

Often times during math we hear students say "Why am I learning this? When will I ever have to use this again" or "in my future career I won't need math". I was without a doubt that person. Little did I know, math is in our every day life whether we realize it or not.

Math... 

is in the food you eat
is found in nature
helps you to build things
is found in the grocery store
... AND SO MUCH MORE. Math is EVERYWHERE. 

When creating math lessons for our students, it is critical that we demonstrate the real life application each topic will have. By doing so, we will create a mindset that math is valuable to our every day experiences. Instead of teaching students only through instruction and worksheets, we need to bring the problems to life. This is so easy to do, and it allows for a rich lesson that students will take more from. It will also likely help them to remember the topic because it was engaging and interactive.

Zehrs. (2016). Zehrs Grocery Store Flyer. (Online Image).
 Retrieved from: https://www.zehrs.ca/print-flyer 

For example, we all know that when we go to the grocery store we use math. Whether it be to calculate how much we can spend, seeing if the sale is a good deal or not, or what change to give to the cashier. During number sense and numeration, we can recreate a grocery store in our classroom allowing students to coin these skills. They can shop throughout with coupons, gain an understanding of how much of an item will equal to the pounds (i.e. when a ticket says $3.45/lb) and creating change to pay for the purchase. By doing so, we give students the skills that they can then transfer to the next time they go grocery shopping.

Another great example is to have students bake something. Through this, they can look at fractions and calculate how many cups or grams they need for each ingredient. During this time, students are participating and measuring the amounts. This provides a great visual to see for example how much one cup will equal.

Get The Math is a great website that draws from student interest to bring math to life. It shows videos from sports, music, video games, fashion, etc. and discusses math components in it followed by a question for students to answer. This highlights a critical step as a teacher- KNOW YOUR STUDENTS. By knowing our students, we can create lessons that are valuable to them and create a better understanding. Growing up I was very involved in sports, and when teachers incorporated hockey into it, I instantly could make the connections and understand what was being discussed. As math is found everywhere, we can be diverse in adding examples to our lessons. Another benefit is bringing cross-curricular understanding to the lesson. Students can further make connections between the subjects through examples. For instance, there are a lot of examples of math in nature. We can easily incorporate that into a science lesson such as the Fibonacci sequence. Another fantastic resource for teachers is Real World Math. On this website are strategies and lesson plans that can incorporate math from every day life into a lesson. In addition to providing real world examples, we can also bring the lesson to life by hearing from people who are in various careers that use math. TED Talks are a great tool to bring to the classroom for students to hear from real experiences.

Math and Technology 

As we working with 21st century learners, it is extremely important that we bring technology into the lesson. Playing various math games, and activities through apps or websites can be incredibly valuable when properly executed. These games need to provide students with a learning opportunity instead of being time fillers. One of the most important things is ensuring that the game is meeting overall, and specific expectations for the given topic. Once we know a game is a rich and meaningful learning experience, it can be incredibly valuable for students. It targets differentiated instruction, allowing all students to participate at their required needs. It also encourages students to have fun, and perhaps even forget they are learning about math.  It is important to also select games that allow for learning such as different levels from beginner to difficult so that students can improve their skills. Games should also be appealing with good content and visuals. There are many apps and games available to play, and incorporating one into your classroom again highlights on the notion of knowing your students by incorporating games that are relevant to their needs.

Smart Kids Software (2016). Math Journey Cover. (Online Image)
Retrieved from: https://www.smartkidssoftware.com/ndlec38.htm 

When I was in elementary school, I remember playing this game by Reader Rabbit called Math Journey. For the time, it was very appealing to users. It had a great story line, and visuals as well as being exceptionally interactive. I can honestly look back and say a lot of what I learned about different topics like multiplication, fractions, money, etc. were from using that game. It made math fun for me, and made me want to continue learning. When teaching, I will keep in mind that an interactive game, or a real life application connection can have a rich and valuable impact on a student's understanding of the topic, and I will use many opportunities to implement both of these into a lesson.

Mathematics Week 5

Key Principles in Math

TED Talks (2017). Math Cartoon. (Online Image)
https://www.ted.com/topics/math

Recently through modules focusing on Dr. Boaler's work regarding math, my mindset towards the subject has shifted. She discusses that because one has been taught math the wrong way, or has a negative mindset towards the subject it affects their ability to understand just how simple it can be. I used to think with math, I wouldn't do well because I either get the answer right or I get it wrong. Vividly I remember during university math subject related exams experiencing panic attacks because I had little to no answer on a question. For example, during biomechanics I can remember having a huge question worth 40 marks and if I didn't get one part correct, it would affect my ability to score any marks throughout the rest of the question. This made math scary for me, and extremely difficult. I felt I had to memorize so many steps, and formulas in order to solve a question that I didn't really think about the good math offers.

Math does not have to be complex, like I had originally made it to be. If you understand the key principles behind math, it makes it far less difficult than anticipated.  Not only that but it allows a student to enjoy the subject, without stress or fear.

Making Sense & Intuition

Arguably the most important stage is developing a sense of math. Teachers have a huge impact on a student's ability to develop sense of math, like whether it is a positive or a negative mindset towards the subject. However, it largely depends on the student's own autonomy to make sense of math. When introducing math to students and more so difficult topics, it is important that educators engrain the development of intuition and sense of math. Through intuition, math makes sense. Students need to ask questions like How does this work? What other connections can I make to this problem? How can I break this problem down? By encouraging the questioning process, we are allowing students to gain their intuition, and become confident in answering the question. How many times have you looked at a math problem and immediately thought I can't solve this or I have no idea how to do this. With that negative response or fixed mindset, we are limiting abilities. Instead through questioning, students can gain the ability to manipulate the question, even if it is difficult to understand at first. In the video below, Sebastian Thrun highlights the importance of allowing time to solve for a question, and building the intuition and understanding. He states that you can't make any progress without really understanding. I thought this was an interesting connection to how math constantly seems rushed. Whereas if we give students the time to build their intuition, we empower students to remove the fear of math, and solving problems.

The Importance of Drawing and Representing 

Drawing and representing has a huge influence in a student's ability to understand a problem. When we provide visual representation, it brings the problem to life and provides another avenue to approach the problem. This is especially apparent when looking at learning styles or preferences. When a student creates a visual or is provided a visual they can have a better understanding of the problem. 

Here's a great example of an alternative visual way to solve for algebra that differs from the FOIL method: 

Swayze, R. (2017) Algebra Visual Graphic. (Personal Photo) Retrieved from Personal Library 

Ideas or Memorization? 

The focus of math is to always think about the idea, and to understand math means you understand the Big Idea. A Misconception in math is that you need to memorize formulas and steps. However, the key principles demonstrate that through drawing and meaningful representation and intuition, you can understand the Big Idea without having to memorize. If we focus on memorizing formulas and steps, we are limiting our students to understand the Big Idea of math, and therefore limiting the opportunity explore a subject that provides a wealth of knowledge and skills for the every day world.

Mathematics Week 4

Ways to Improve Mathematical Instruction 

LRQ Consulting (2017).Flexible Pencil. (Online Image).
 Retrieved from:
 http://www.lrqconsulting.com/2017/02/13/grow-with-flexibility/

Math Flexibility 

The common view of mathematics is a negative one, at least within majority of the people I associate with. Math was always a concept that people felt there was an absurd amount of rules and methods that one had to memorize, that there's only one right answer, and that there's no room for creativity. I very much believed that as well, until now. Math has continued to be a subject where I have had many negative feelings towards from grade school to calculus in university. I could understand it, but it required a lot of patience, and coaching myself to get through it. However, recently I've learned that math can be quite enjoyable when taught in a way that targets the mindset of the intuition and creativity. I have realized that math does not have just one answer. For example, when looking at math flexibility, we may reach the same answer but the way we approach it can be quite different.

With my fellow classmates, we were asked to solve 18 x 5 using mental math. I approached this question by breaking it down. I first solved 10 x 5 to get the answer of 50, and then solved 8 x 5 to get 40, where I then added 40 + 50 to get the final answer of 90. 18 x 5 = 90. I noticed some peers solved it in a different way, but reached the same answer. For example some took 20 x 5 = 100 and then took 20-18 = 2 and multiple 2 x 5 to get 10. Where they then took 100-10 to get the answer of 90 for 18 x 5. I thought it was incredibly interesting how people solved for this. When I was in school, we were literally taught only one way to solve it. Now my knowledge and skills and strategies are expanding through seeing alternative ways to approach a question. In future classes I think it is hugely valuable to embrace math flexibility because we can grow our "Math Skills Tool Box" in discussing different approaches to reaching answer.

Dr. Boaler discusses that people dislike math or do not do well in math because people were not introduced to math in the right way which leads to a negative perception or a fixed mindset. With that, if we create a growth mindset and incorporate flexibility we allow students to discover new ideas, respond to answers creatively, and truly learn and understand the mathematical processes. Through incorporating flexibility, we can allow students to think outside the box, and gain an appreciation for the subject because their confidence is being built.

Rich Tasks 

"Rich tasks encourage children to think creatively, work logically, communicate ideas, synthesize their results, analyze different viewpoints, look for commonalities and evaluate findings" (Piggott, 2011). 

This past week I was introduced to the concept of Rich Tasks, and I feel it's an incredibly valuable topic to share with fellow educators to implement into their own lesson planning. View the Rich Tasks Chart to see the contributions it can make towards the lesson. We had an opportunity to solve the Finger Counting Problem which was a great first hand experience at understanding the value rich tasks can bring to the classroom. 

Here's how my group solved the problem: 

Swayze, R. (2017) Finger Counting Problem.
 (Personal Photo) Retrieved from Personal Library 

                                   

Rich tasks have the ability to encourage math flexibility. As the entire class participated in this open-ended question, students were able to solve the problem in a way meaningful to the group. Upon completion, as groups presented the way they solved for it, it was amazing to see how the flexibility was at play with everyone leading to the same answer a different way.

The article, How to Help Students Develop Flexibility in Math, provides great strategies to implement in your classroom that encourages flexibility practices. This includes building a culture of sharing, and encouraging students to take more than one approach at solving a question.

Check out Which One Doesn't Belong for an awesome resource to begin the Minds On portion of any math lesson!

Quotlr. (2017) Albert Einstein Quote.
(Online Image). Retrieved from:  https://quotlr.com/image/2968 

                 

Mathematics Week 3

Learning Styles... Preferences 

Mercedes (2011). Left vs Right Brain (Online Image).
 Retrieved from: https://adsoftheworld.com/media/print/
mercedes_left_brain_right_brain_paint

I had an "ah ha" moment in class regarding learning styles. We have had many discussions on reaching all learners needs by providing different styles of instruction that adhere to the student's individual learning style. However, through discussion in multiple classes this week, I truly realized although extremely important, it is a preference. Too often we hear someone say, "oh, I don't understand, I'm a visual learner", which is likely true but they come to depend on that style without learning through the rest. I consider myself to be a kinaesthetic, tactile learner and I find I do best when I learn in that way, and I have sort of become dependant on that. There are definitely times when someone is giving me instruction or directions through an auditory style, and I honestly blank on the information. My mind is distracted thinking about other things, and I panic midway knowing there is no way I am going to retain this information- simply because "I'm not an auditory learner". After thinking about this notion, I think it relates to the concept of growth mindset. By solely relying on one style of learning we are limiting our ability to learn information, and brain growth... perpetuating the fixed mindset. I want to provide all students a style that makes them comfortable but I also see the importance of challenging them in other learning styles, and increasing their growth and ability to retain information. Through using a variety of learning styles, we can increase the knowledge, and even student interest. We allow students to use both sides of their brains equally creating imagination, creativity, analytical, and logical thinkers. Through challenging themselves in different learning styles, we can encourage a growth mindset and give students the confidence and motivation to understand information that is coming at them in different styles.

Differentiate Instruction

To foster this growth mindset, it is important to differentiate instruction. This can be completed through:

1. Content: Delivering curriculum knowledge and skills through a variety of methods including: video, images, reading, instruction, audio

2. Process:  Making sense of the content through reflection and digesting learning activities such as:  think-pair-share, journalling, partner and group discussion, games, acting it out, media literacy

3. Product: Learning artifacts and evidence of learning including: provide students choice on how they want to present their knowledge- i.e. a PowerPoint, a skit, presentation, journal, etc.

Mistakes... 

Technology Rocks (2012).
Mistakes Quote. (Online Image)
Retrieved from: http://www.technologyrocksseriously.com

Did you know your brain actually grows when you make a mistake?! I had no idea. Once I learned this scientific information, I thought why are mistakes considered so negative if they have such a positive effect on the body? When a mistake occurs, the brain struggles to find an answer which in turn strengthens brain neurons. This can be compared to exercising any other muscle in the body, the more you work it, the more the muscle strengthens. Mistakes, and taking time to solve a problem need to be emphasized in the classroom as a positive experience, not a negative. Too often, students are rushed to find an answer and when its incorrect a simple X is placed not allowing the student to grow and learn from it. It's difficult, there's only so many hours in a day, however, we are truly not allowing our students to grow if we do not give them the time they need to solve problems and learn from their mistakes. Through emphasizing mistakes, we can foster a growth mindset, encourage creativity, and provide valuable life lessons when experiencing mistakes. As this notion is becoming more aware through research, education is making math problems longer and more open ended which allows students to struggle and challenge themselves. With that, we allow students to take their time solving which provides students the opportunity to invest in deeper thinking. If we place time constraints, we are having a negative contribution on students learning and their ability to excel in math.

A wonderful way to encourage deeper thinking in math and giving students the opportunity to learn through their mistakes is through Cathy Fosnot's method of Strings. During this time, students are presented with a set of questions related to unit topic, in which they have time to solve and strategize to find the answer using mental math and not a calculator. Afterwards once each student has had the opportunity to find an answer, the class returns to discuss and see the many different ways the problem can be solved. It's so cruical that we take the time to give students the opportunity to invest in deeper thinking as the benefits far outweigh any negatives. 

ResumeWriter (2015)Albert Eistein Quote. (Online Image).
Retrieved from: https://imgur.com/gallery/UlvVXAC

                 

Mathematics Week 2

Math Mindsets 

Class teaching- Find the bright spots (September 2013)
Growth Ahead. (Online Image) https://classteaching.wordpress.com

Is it a math mindset or is it a life mindset? When looking at the terms of growth mindset and fixed mindset, it's a life style choice that applies to all subjects, and how we carry out our every day tasks. We're currently living in a world, where you see more negativity in the news than positive acts. The Generation Z is living in a world surrounded by technology, social media, and always being connected. A world that is constantly posting their every move, and looking for the next like. This itself creates a fixed mindset. The want to stay in their comfort zone, the fear of making a mistake, and the want to look smart, and popular amongst peers.

How can we implement a growth mindset in the classroom, if the rest of the world isn't truly encouraging growth?... That right there is a fixed mindset. As educators, we need to encourage our students that anything is truly possible whether it is a big impact or a small impact. The brain is a muscle in the body that just like the arms or legs, and needs to be conditioned into being in it's best state. The idea of growth mindset will not happen over night, and there will be setbacks but through practice and perseverance we can create this mindset. As a child develops so too will their growth mindset.

This positive development needs to be practiced each and every day to create a life mindset that students will take with them throughout their lives. Subjects that students learn throughout elementary school are each a piece of a large puzzle which is the development of the whole individual- educational, social, and emotional. Often times, math is the subject that has the most fixed mindset. If students have a fixed mindset in math, how will that carry throughout their life, and the mindset they have later on?

With that, it is critical that we as educators bring a positive light to mathematics, recognizing that it is difficult and challenging, but not impossible. To do so, we must recognize the language we use in the classroom, our attitudes towards the subject, and how we present growth mindset strategies in the classroom.

Carol Dweck, a leading psychologist from Stanford formed the growth mindset, and states "there is no relation between students' abilities or intelligence and the development of mastery-oriented qualities. Some of the very brightest students avoid challenges, dislike effort, and wilt in the face of difficulty. And some the less bright students are real go-getters, thriving on challenge, permitting intensely when things get difficult, and accomplishing more than you expected". I think we can all relate to that research in many aspects of life whether it be education, sport, music, etc. The people who fight for it, and challenge themselves despite abilities are often the more successful ones.

In future classrooms, I want to see students challenge themselves, preserving through the difficult stages, and accomplishing anything they choose. With that its important to have strategies, and demonstrate growth mindset throughout the classroom, and guide student's to develop their personal growth mindset.

Strategies: 

When developing a growth mindset it is important to be aware of what your student's like, and what will encourage their growth mindset. Showing videos, such as this series guide visual learners to their growth development and to the 21st century learner, it is more enticing.

Mashup Math. (2016).
Promote a Growth Mindset.
(Online Image)
http://mashupmath.com

As math can be a nightmare for most students, it's important to implement strategies specifically targeted at the growth mindset of math. In Cultivating a growth mindset in mathematics the article discusses strategies formed by Jo Boaler that help encourage a positive growth mindset in mathematics. As a leading researcher, her goal is to demonstrate how beautiful math can be, and allow student's to explore math with some freedom. This includes reduces the test-taking pressures, create a learning subject rather than a performance subject, and adapting math to today's technology world.

As educators, we need to constantly research how to better our classrooms, and there are a wealth of ideas regarding growth mindset such as EdSurge's excellent article 4 Ways to Encourage a Growth Mindset in the Classroom . The development of our student's growth mindset begins with us, and how we conduct our classroom each day through the language we use, and how we encourage our student's to overcome challenges. It is critical that we recognize that math is challenging to many people, but through challenging yourself, and practicing, you can improve. Relating to other aspects of student life such as their interests can help student's to find determination. For example, if a student really enjoys hockey discuss how it was challenging at first for them to learn to skate but through practising, they were able to improve, and the same can happen with math.

Mathematics Week 1

The Power of Yet Challenging Math Myths and Stereotypes 

Wikipedia (2017). Four Queens. (Online Image).
https://en.wikipedia.org/wiki/Queen_(playing_card)

During our last class, our teacher started with an excellent activity that demonstrated the power of words we use as educators. In this activity we used cards where a student would split the deck in half, and then split the two new decks in half again. Once that was completed, three cards would be taken from the top and put on the bottom. The dealer would then place 1 card from that pile on the three additional piles. This pattern continued three times, where once flipped it was discovered that all four queens were on top!

We then split into groups to attempt to figure out how to do the trick. Within my group, three of the four of us proclaimed the classic line "I'm not a math person". I felt this was interesting that before even trying the task, we felt the need to state our positions on math.  On the first try, my group had 2 of the four queens. We thought we were doing a great job at figuring this out, and that it had to be passed on patterning and which card goes on top. After a lot of frustrating effort, we later realized we were tricked and that when beginning this activity you have to ensure you know where the 4 queens are. One key thing our teacher had said in the beginning was "this is easy", and throughout the activity I thought how in the world does anyone think this is easy.

With this example, we were taught the power of our words and vocabulary towards our students. By saying an activity is easy, it can cause stress and reducing confidence. Instead we need to emphasize the power of yet, and how it may be difficult now but thorough practice we can improve!




Game About Squares 

Swayze, R. (2017) Game About Squares. (Online Image) 

We had the opportunity to incorporate an activity that taps into the 21st century learner. Game About Squares is challenging, and without instruction allow students to explore inquiry. It promotes critical thinking skills, and emphasizes the importance of not giving up. An act that many do in math. This can be a great tool to use on devices during transition.











Brain Growth

Throughout this video, Jo Boaler highlights that every individual can learn math. There is no such thing as a math person! She also emphasizes the importance of mistakes in improving your abilities. And one of the biggest take home message is the attitude and belief we have towards math influences our ability. With this research, it is important as educators to have a positive attitude around math, embrace that all of our students can do math, and incorporate a growth mindset.

Going into the new school year, this is an area of focus that I will be looking at to improve my growth mindset on math. As educators it starts with us, and if we do not display a positive and open attitude towards math with our students, we cannot possibly expect them to do the same. I look forward to exploring ways that we can engage all students, and ways to make every learner successful!