Reflection

Over the past month or so I have learned and developed numerous crucial skills that I can use in my placement as well as my future classrooms. I have learned that math has changed drastically from the time I was in elementary and even high school and it has changed for the better. Math is a subject that students no longer need to feel afraid of but rather inspired and a subject that they look forward to. Teaching has switched from direct teacher instruction and memorization of formulas students do not understand to the teacher acting as a guide for the class allowing for student inquiry in any way possible. Students are free to develop the skills they need to problem solve in their every day lives and realize that they use math every day and in almost every situation possible. 

It is important to note that when teachers allow students to participate in student inquiry and student centered learning they are exploring topics to a much deeper level of understanding rather than direct teacher instruction. It is important that students understand mathematical concepts and are not merely memorizing formulas and copying steps down that they do not understand because when it comes to application on future assessments the students tend to not do so well because they are not able to apply the knowledge they have in a proper manner because they did not grasp a deep understanding of the concepts. Allowing students to solve open problems that are found in their every day lives allows them to understand mathematical concepts and also apply them. 

I am to be a math educator that inspires my students to think of math in their every day situations and use it on a regular basis. I want to inspire and allow my future students to grasp concepts through student inquiry and have discussions with their peers on the various methods and approaches one can have to a mathematical problem so that they may understand what approach is more appropriate to use in one situation versus another. Finally I want my future students to be excited about walking into math class and about what they will learn next.

-Miss Proenca

Probability and Estimation

Why do we need estimates? Sometimes we do not want to know the exact answer but instead an estimate is more useful. We make estimates every single day and for many things in life. For example you may estimate how far you can get with the amount of gas in your tank or how long it will take you to get to a location if it is raining. Below is a photo of a jar filled with cookies that demonstrates an activity that is played at many wedding showers, birthday parties, or events that you may participate in life. You can get great at estimation by practicing and developing strategies that work for you.

Growth Mindset:
Our brains are like a muscle and it is important that we can work that muscle and let it grow and develop. Students should never feel like they are not good at math but instead keep trying and practicing and developing their skills because everyone is good at math it takes hard work and dedication. Below is a link to the website named Class Dojo that provides videos made for children explaining concepts such as your brain is a muscle and many more crucial topics.

https://ideas.classdojo.com/i/growth-mindset-1

Data Management and Probability:
Students love yo know what to expect and providing students with parallel tasks allows them to feel like they have some sense of control over their learning and allows them to take responsibility and initiative. Probability can relate to many different aspects of student's lives such as those who are interested in sports statistics. There are numerous games and online activities that allow students to practice probability such as Gizmos Reaction Time Game.

There are numerous data programs that you can find online to include in your lessons and explore data management units with. One of the programs that we explored in class was named Tinkerplots and there are numerous things you can do with inputed data. It is important to not just read the data but read between the data and read beyond the data as well. When teaching data it is not a matter of formulas but understanding the three measures of tendency Mean, Median, and Mode. 

Assessment of Math

Co-operative Groups:
A great way to get students moving around and getting physical such as making shapes with their bodies or attempting to make geometric shapes. It is important to be careful and not make students sit down for large periods of time because they can begin to lose focus and then they are not engaged in their learning. Everyone's voice in the classroom should be heard and there should always be opportunities for every student to contribute therefore provide students with oral clues. Grouping students in pairs is an effective strategy for collaborative work in math and if needed then students can compare their work with another pair beside them. If groups are too large that does not provide fair opportunities for everyone in the group to be a contributing member.

http://mathtechconnections.com/2015/12/24/differentiatedmathgroups/

Clapping Institute:
In class, the activity clapping institute was demonstrated to provide understanding that as educators it is important to give every student clear targets and share learning goals and clear success criteria. As educators we are not simply at the front of the class partaking in direct instruction but our role is to facilitate and guide students in their learning experience. When students are problem solving it is a great opportunity to assess their knowledge and application. Involving students in their own formative assessment allows them to set personal goals, opportunities to try again, and the result is that learning happens.

"Dont' Make Careless Mistakes"
Never tell students to not make careless mistakes. Mistakes are a part of the learning process and students do not mean to make careless mistakes it can be a cruel thing to say that can discourage students from progressing.

Keep Learning Goals Simple:
A great way of getting students to understand the learning goals is to collaborate with students to develop the success criteria. Monitor students progress and adjust instruction when appropriate. Not all teaching strategies are going to work out and keeping track and reflecting on lessons allows educators to make adjustments in the future based on what worked and what did not. It is important to have discussions with students to see what strategies are working best for them based on their learning styles and needs.

Measurement Math Activity

In this weeks class I had the pleasure of demonstrating a grade four or five math activity that could be used in the class to teach the unit of measurement to students. My classmates acted in the role of a grade four or five class and played the activity name banners. The idea of the game is to introduce the concepts of area and perimeters of simple shapes such as squares. Using 1cm squared graph paper the students were asked to write their names down and then calculate the area and perimeter of their name. In the photo below is a demonstration of a name banner that a student may create. The name banner activity is a great way to get students interested and engaged in learning about perimeter and area considering students love activities that are personal or individualized.



Another concept that we discussed in class is that when students can keep track of everything they've tried, they feel like they are making progress. Using charts to keep track of what students have done is a great way to track progress. It is important to reward hard work because the more time students spend on a task the better they get at it. Students must learn to establish a math mindset. Guided Inquiry Lessons are great because they are inquiry based and develop challenging concepts. It is important that what you are teaching is right for the grade level. Read expectations before and up to your grade level to know what they have learned in previous grades. The Minds On portion can be something that is considered review for the students to activate prior knowledge however the lesson should be something new. Consolidation is an important part of the lesson that allows students to discuss what they have learned in the lesson.

 The photo above depicts an activity that allowed students to learn the unit of measurement in groups and not only did it relate to real life but it was also a hands on activity that students could physically grasp the ideas of measurement. Students were able to calculate surface area using the toilet paper tubes and rulers and make connections between the surface area of various shapes.

The Greedy Triangle

What does Similar Mean in Math?

• Same shape
• Could be different size, colour etc. the colour does not affect what is happening mathematically

Teachers can use the smart board as an effective tool to teach Geometry for example using the smart board to demonstrate shapes that are congruent or similar. The smart board can be a great interactive way to demonstrate these shapes and have students come up to the board and rotate and move the shapes around to examine the relationships between the shapes. Mathematical vocabulary is a great way to start a class and ensure that the students are understanding the concepts before solving problems.

With regards to the topic of symmetry it is important to use real-life examples so that students understand what symmetry means. On both sides of the line of symmetry the shape is equal. Students can also use the smart board to understand that if you fold along the line of symmetry both of the sides are equal.

Using Children's Literature to Teach Math Concepts






There are many examples of children's literature that can be used to teach math concepts to students such as The Greedy Triangle by Marilyn Burns or her other novel named Spaghetti and Meatballs For All A Mathematical Story to teach perimeter and area. Reading to students aloud allows to involve them and make it engaging. It allows you to introduce topics and everyone enjoys being read to even older students.

Development of geometric thinking is key to the success of students in Geometry. It is important that students have a spatial experience and that problems include every day life scenarios. The best way for students to really understand Geometry is for them to be touching it, moving it, and cutting it. When students can physically experience geometry they are able to clearly understand the concepts. In Geometry all students should be using manipulatives because it develops a different area of the brain that is crucial for understanding. 

Mistakes Are Important

In math mistakes are an important part of success. Teachers should move away from the more traditional approach of teaching where students are afraid to make mistakes and fear asking questions because they do not want to look "foolish". Well questions and mistakes should be encouraged because that is an important stepping stone in learning. It is important that teachers change the way we encourage students and reward their efforts and hard work by saying compliments such as "You worked so hard, I am so proud of you" as opposed to "You're so smart" An effective video for demonstrating this concept is the Meet the Robinsons, You Failed! video available off of Youtube.




As educators it is important to use letters in equations that apply to the questions being asked. Instead of using X and Y because how many words really start with X and Y? We can use letters such as b for blocks or c for cats and so on. Students start to really understand proportional reasoning when they look at patterns with scenarios and examine the relationships in these problems. Incorporating the use of the smart board in math can also be useful for students to get engaged in their learning. For example one can explain mathematical vocabulary such as output, constant, and variable with visuals on the smart board such as demonstrated in the photo below.

Teaching students concepts such as what is increasing by the same amount each time? And therefore that would be your constant is an easier way of understanding functions than simply memorizing formulas and equations because then students have a much deeper understanding of equations and how to formulate one themselves.

Deducing Problems In A Math Congress

By participating in a math congress the student role becomes much more active as students take part in a congress where they can discover how other students solved the same problem and ask questions about understanding the various ways that a student can choose to arrive at a solution. A math congress demonstrates a growth mindset where dedication and hard work are depicted in the student's solutions. Students can greatly benefit from understanding that what they bring to the solution is valued and that the teacher does believe in them. 

One of the most important goals is feedback. Once a student can see and understand a problem then they can go further on their path of solving the problem. Often another student may explain to that student where they went wrong or help them to further understand the problem so it is easier to solve. The most important part about solving problems is taking what you already know and applying that information accordingly. Making connections is a crucial step in problem solving. It is important that as educators we provide students with rich problems to support their learning. 

Different groups of students approach problems in different ways and all of them can come to the same solution. In the case of Joel's Kitten Food Problem, some groups chose to focus on the dollar value or the can value, however way the students decided to approach the problem they are still able to come up with the right solution. 

Math congresses allow students to understand every solution and make connections between them. A math congress is student centered and students are able to find out what information is new to them and why they might be able to use one approach versus another approach at different times depending on the problem at hand. A math congress gives students ownership of their own work. Students work better when they know their work and method is valuable and when they have control or a choice. When students make sense of a problem they can solve it. Math congresses allow students to be valuable members of a group and take responsibility; both characteristics that are valuable to their futures. 

For The Love of Fractions

When teaching students about fractions it is important to not revert back to the old methods that were taught to us in the more traditional approach such as cross-multiplication. Students have no idea why they are cross-multiplying and really do not understand fractions entirely with this approach. Instead it is crucial to provide students with good open math problems involving fractions to deepen their understanding of fractions instead of memorizing formulas. Why not simplify dividing fractions for students and let them divide the tops from the bottoms. That is what makes sense and works better for them. I feel that if a teacher in my past had told me it would be okay to divide fractions this way that I as a student would not have struggled or been as frustrated with fractions as I had been. I feel it is important to keep math simple and let the student understand the big ideas so that they can progress in their learning experience and not feel discouraged.

Parallel tasks in the classroom are essential to promote growth in a student's learning experience. Parallel tasks include more challenging or less challenging tasks where students can choose which one they may want to pursue, and then at the end the whole class can participate in a discussion. It is important to not give students generally more than three choices so they do not spend the majority of the time deciding which question to do but stay on task. Students may have off days where they do not feel as great or confident as other days and may want to pursue a less challenging task or students may want to challenge themselves further by picking a more challenging task that day. Whatever the case may be, the student can always pick a question that they can get started on and discuss with their peers.

There are fail safe strategies for creating open questions for students such as begin with the answer to a question and allow students to work on the problem to arrive at the solution. Another strategy used is asking students for similarities or to leave certain information out of the problem and allow them to fill in the blanks themselves. It is crucial to allow students to be scientists and allow them to to try methods out by trial and error. Students can compare answers to see if they are going in the right direction or what trajectory they should take. Fractions can be applied to other units taught in math and you can teach lessons that tie together themes so that students can make connections between the lessons. You might then hear a student say " Oh yeah, I've seen this before. So ratios and fractions are pretty much the same thing!"

Fractions or Ratios or Proportions? How About All Three

There are so many ways of looking at fractions such as it can be comparing different parts of something or parts of a whole. It's important to understand that ratios and proportions are just the same as fractions. There are various types of fractions such as proper or improper, mixed numbers, as well as unit fractions. The most important thing to remember is that coming up with a good problem means getting students to bring up the important vocabulary surrounding fractions such as numerators, denominators, and what fractions mean to them as students. Math is found in our every day life and more specifically we are surrounded by fractions. Getting students to demonstrate fractions in their own ways using manipulatives of all kinds can provide them with a visual representation. A good problem is one that has a wide base where every student can get started and the problem that was demonstrated did just that. Students were able to pick their favourite fraction and represent it in as many ways possible without any rules or regulations. The only drawback to this problem is that there was no real life context provided that can always be a challenge when introducing strands of math.

I thoroughly enjoyed the segment on teaching through children's literature because it proved how math can be related to other subjects as well as how fun it can be for students. The Hershey's Milk Chocolate Fractions Book provides a fun and engaging way for students to be introduced to important vocabulary with regards to fractions as well as an introduction to the strand. What student wouldn't want to learn math with chocolate? It is crucial that fearless speaking and listening is encouraged in the classroom to maximize student's learning. 

A growth mindset is important with regards to teaching math. Providing students with parallel problems gives students with a broad range of learning styles and student readiness to get started on the questions that they want to. It is important to spark up reflection and discussion regarding the problems. Questioning is one of the hardest parts about being a teacher and its important not to give out hints to the student who is asking for help but rather asking questions about connections they have made or getting them to communicate about what they have done so far and what further steps to take.

The Big Ideas from Dr. Small book is a great teacher resource to use. The book looks at the big ideas for every topic taught in school. So often, the specifics of a topic such as math are taught and students tend not to understand the big concepts. The key is to get students to understand the big ideas or larger concepts and have a deep understanding about what they are doing in math and why they are doing it that way rather than just memorizing formulas and applying them without any further reflection.

Algorithm Galore

In this weeks class we learned about the importance of different algorithms with respect to math. It is important to get a rational understanding of why we got the answer we did as well as how we got the answer we did.  The assessment for, as, and of learning are crucial to see and understand how students learn. All students don't learn in the same way and it is important to get students collaborating and discussing how they think in order to appeal to different learning styles and techniques. Students can understand that there are different methods to solving a problem and that the method they may use in one situation varies from another. Students understand that math is meaningful and it is everywhere. Getting students excited and involved in their math experience means that it becomes more enjoyable and students begin to love math rather than dread coming to class every day. It is important to note that students learn from each other and in various ways when teaching students math content and how to solve problems.

The images above demonstrate that there are many ways to solve a problem in math and some students may understand how to get it right away while others are more visual learners and need to break down the problem visually. The second photo really appealed to me because I myself am a visual learner and I can begin to see how a student will understand what multiplication really means by seeing how it is broken down into parts. Students should not cry while filling out multiplication tables at home because they are struggling to complete them but rather use multiplication as a simplification tool just so they don't get bogged down in numbers. Multiplication has many meanings and it can be thought of as repeated addition, counting of equal groups, or objects in array. Thinking of multiplication as an area of a rectangle is a brilliant way to visualize a concept that many students may just think of as numbers at first. It is important to understand the meaning of multiplication before they simply just memorize a table and move on.

Students need to see math as sensible and useful in their daily lives. Good math problems involve relatable material such as holidays, sports, and material they are interested in such as super heroes or pokemon. A good math problem is one where any student can get started regardless of their comfort level in math and is something they can expand or build upon. Good math problems often have more than one answer or at least multiples ways of achieving a solution. Open problems are the key to any student getting involved in math and interested in developing their math skills

Retrieved from www.giphy.com

Relatable Math? Yes Please

So often we hear negative messages about math such as “ I hate math” “I’m not good at math” or “I’m just not a math person”. Even in Hollywood movies we get bombarded with how math is for “nerds” or girls just are not good at math. Week one has completely changed my perspective about math and how fun it can be. It is so useful in our daily lives and I think the old methods of teaching made us forget about that. The old school methods of teaching created an environment where students were afraid to ask questions so they wouldn’t seem “stupid” or feel devalued if they just couldn’t understand the complicated formulas that were thrown our way. Teachers worked at paces that were too fast for students and complicated things way more than they needed to be. Why should there only be one way to solve a problem? Students should be able to solve problems in their own way and not think that there is only one “right” way of solving a problem.

           

            I was mind blown by the example of how to add and subtract that was demonstrated in class. I had always practiced addition and subtraction the way that my teachers had taught me previously but I never really understood what I was doing or even questioned it. Why would students be borrowing numbers from the column beside? When you borrow a number, where does it go? And why complicate math on paper when you can just simply do what is natural and the way that we all do math in our minds? I couldn’t believe how much more simple the way that was shown in the photograph above is. Students can easily add and not only that they understand what they are doing. I love that I can embrace that there are different ways of answering a problem and that everyone can come up with their own way that they feel comfortable with just as long as the method is not specific to one question and can be applied more than once.

            I am looking forward to learning other new ways of solving math that makes 
it easier for students to understand so that they never have to feel like they hate math. One message that really stuck with me this class was that a student should never say that they just can’t do it or they don’t get it because they may not get it in that moment but with the use of manipulatives and asking further questions or creating real life situations they will get it, they just haven’t got it yet. I love that math can also be relative to the current life of students. Math problems should not consist of “farmer brown has 5 pigs”… but have to do with fun themes that students can relate to. Math problems are open ended and get students engaged and asking questions that spark their creativity and they are able to use the knowledge they already have to solve the problem. All students have strengths and they can finally demonstrate those strengths and not feel ashamed about their weaknesses in math. 

Retrieved from imgur.com