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