Weekly Reflection 4

 Hi everyone and welcome back to my blog!

Today I wanted to discuss the importance of thinking of math in non-mathematical ways – this can be a very valuable strategy to students of all ages. Math tends to hold an image that only linear thinkers can understand and accel at it; however, this is not true at all. It is important for educators not to create an atmosphere that believes this, as it will result in a fixed mindset. Anyone can accel at math, and there are various ways to understanding math without only focussing on numbers and formulas. Having students first look at problems in a way that is more visual may be a necessary step for many students before they can arrive at understanding a complex math formula. 

The Dot Problem 

            The Dot Problem is a good example of this – instead of having students come up with the formula that represents the growth pattern, have them take the time to see the pattern in their own unique way and describe it through images, words, numbers and personal connections. This way, students can construct a formula that makes sense to them. While this way of arriving at an answer may take longer, this level of learning is much more meaningful. 

Some ideas generated from students in class

When discussing this in class, we were told to ignore our innate desire to create a formula right away. Instead, we were to represent the pattern using images, words, manipulative and making connections wot what it represented to us. Once we each had our way of seeing the pattern, it made it easier to visualize what changes were happening at each stage. If looking at the image growing by 4 corners each stage, we could represent that as +1 +1 +1 +1 on each corner. Breaking the problem down into manageable pieces made it easier to construct, and actually make sense of the expression.

The representing Process Expectation for grade 4 students is to "create a variety of representations of mathematical ideas (e.g.,by using physical models, pictures, numbers, variables, diagrams, graphs, onscreen dynamic representations), make connections among them, and apply them to solve problems" (pg. 65) which applies directly to this method of solving the Dot Problem. While constructing the formula and getting to the final destination is obviously important, the ways we plan, organize, connect and represent are just as valuable. 

The Power of Math Manipulatives 

            

            Additionally, I wanted to discuss the importance of math manipulatives. While any opportunity for hands on learning is beneficial for students, some manipulatives are better than others, depending on the problem. It is the teacher’s responsibility to assess the various forms of manipulatives, and offer ones that benefit student learning, not inhibit or restrict them.

Christine used marshmallows as her manipulative!

For example, in a lesson about geometric shapes, Christine used toothpicks and marshmallows as her manipulatives. Not only were they very engaging for students of all ages, but they also offered flexibility, allowing students to explore geometric shapes creatively, with no limitations. In contrast, while Maxwell’s lesson was clear and helpful, his choice of linking cubes were not the appropriate choice, because they only allowed students to explore rectangular prisms, and that is it. I found it very frustrating with these limitations placed on what shapes I could explore and what ones I was unable to; thus, considering how beneficial the manipulatives are to student learning is essential before assuming that they will automatically help because they are a manipulative.

The math textbook suggests many math manipulatives

As shown here, the math textbook offers numerous manipulatives that are beneficial for student learning and engagement. However, the teacher should be cognizant of choosing an appropriate manipulative, so they better fit the activity and improve student learning. 

Thanks everyone for checking out my blog!

– Madeleine 

Weekly Reflection 3

Hi everyone!

Thanks for checking back in!

The Importance of Group Collaboration in Math

This week I really want to discuss the importance of group collaboration within math; I've come to the realization of the importance that group discussion has, especially when exploring new and unfamiliar areas of math. Sure, working in isolation is important too, and completing worksheets and practicing questions will help refine individual math skills; however, when learning a new unit, topic or problem, I think starting off with group collaboration is really beneficial for the students. This allows students to communicate with other students, hear their ideas, learn to problem solve and approach the given issue in new ways – as well, group collaboration helps you to push yourself towards success, possibly more so than problem solving alone offers.

Daniel's integer activity 

This was exemplified in Daniel's math activity – he had us playing a fun and challenging game to find out the missing integers with only 4 clues. At first, our group worked individually about the problem – to be honest, I copied what my group members had down but didn't necessarily understand the reasoning as to why. This method took us awhile to solve the puzzle – I was very confused, felt like I was adding no value to the group dynamic and wanted to give up (in fact, I did give up). However, for the next problem, we read the clues together, communicated with one another and took the time to explain our reasoning and suggestions. With this method, I found myself more engaged, excited to solve the problem, understanding the task better and actually felt as if I were a contributing team member! We solved this problem MUCH quicker than the first time – of course some of this had to do with the fact that we were more familiar with the task, but I believe that a lot of it had to do with the fact we were working as a team.

Danielle's Fraction activity 

 If I had to do this activity alone, I'm fairly confident I would not have been able to solve the puzzle. Reflecting on this, I really want to incorporate the element of group work when I can within my math lessons – I was surprised at what we were capable of as a group. As a teacher, I need to remember that the students are capable of solving these math problems, if given the time and resources to do so. A learning moment happened during Danielle's presentation when she gave us the formulas to remind us of how to divide/multiply fractions. I may not have remembered or known what to do for the activity without this cue if I were doing it alone; however, with the help of my group members I'm sure we would have been able to figure it out. We reflected on this in class – while the purpose of an educator is to help students, we often forget that students are able to teach themselves without us, and we should act more as an aid than a crutch. If we give our students the time to work in a group before we give them a solution or formula, the learning that happens is so much more valuable.

In my placement, my Associate Teacher has her students sitting in groups, and she often allows them time to break off into these seating groups and discuss the problem given. Walking around, I am able to listen to the conversations happening, the students teaching one another and the communication that is happening. My Associate Teacher also tells her students that they should first reread the question, then ask a partner, then ask a group member and then finally, if they are still confused they can put up their hand to ask her. Her students respect this motto, and I personally really like to see the students working together to find out how to understand a problem before running to the teacher for help.

This method of learning also poses an opportunity for discussion and reflection within the group setting. A process expectation (reflecting) for students in grade 8 math is to be able to demonstrate that they are "reflecting on and monitoring their thinking to help clarify their understanding as they complete an investigation or solve a problem" (110). In order to achieve self reflection, I think students have to first learn to reflect with one another when solving problems. This also teaches valuable communication and cooperation skills, not just problem solving. 

Thanks for reading guys! I'll check back in a couple of weeks!

Madeleine 

References:

Heiman, Nikki (2014). Role playing equals an effective strategy for group work. Retrieved from https://heart2heartteaching.com/role-playing-can-be-effective-strategy/

Ontario Ministry of Education (2005). The Ontario curriculum grades 1-8 mathematics (Revised). Toronto, Canada: Queen's Printer. 

Weekly Reflection 2

Hey everyone! Welcome back to my math blog. Today I'm going to be updating you on some material covered in class, and how I will be utilizing elements in my classroom. 

These past few weeks have been really helpful seeing my colleagues present their math lessons to the class – I’ve grasped a good idea from what is expected from those who have done well, to what I would need to improve upon for those who missed vital criteria from the rubric. I think most activities focused on helping students to visually see the representation between numbers; as I am a visual learner myself, I feel like I will address difficult areas of math (such as fractions and decimals) to my students firstly through visually understanding them as numbers, in comparison to their whole. In math class, I was always a student who wanted to know “Why?” and found it frustrating when I was understanding a concept merely as just a formula, unconnected and isolated from its content. Having students first understand and see fractions compared to their whole numbers, I will then help them to make relationships between the two, as a way to connect and give fractions meaning.

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 Personally, I really enjoyed my colleague Kyle’s presentation – he was organized, his activity sheet was clear and concise, and his activity would have helped me (if I were a grade 4 student) understand relationships between fractions and decimals in a fun way. Combining elements of game and challenge (memorizing where each card was) as a way to introduce fractions and decimals was very smart and would be an engaging activity I would use in my class. Many students are visual and kinaesthetic learners, so after explaining the rules and the objectives for those auditory learners, I would then let students have freedom to explore fractions and decimals, in a more tactile approach. For Kyle's game (pictured here) we got into groups and played concentration, while attempting to match the decimal representation of a number with its matching fraction representation. He supplied us with an answer sheet, base-ten blocks, as well as an opportunity to make our own cards! If I were a grade 4 student, this would have been a fun, challenging game that would help to memorize basic fractions and their decimals. 

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          Another activity I enjoyed was Colin’s presentation – he was very comfortable talking to the class and was very easy to understand. While I had already learned his ‘9 times-table trick,’ I would still have been able to clearly follow along had I no prior experience. This is a very useful tool to teach children, as the 9 times-table is very difficult to comprehend. His use of both manipulatives, as well as the physical hand, applied to various aspects of learning.

This is an image of the manipulatives that Colin created – while you had the option to use your hand, these hands were numbered to help make visualizing the problem easier!

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           Another helpful resource I will use in my future classroom is from the textbook Making Math Meaningful– while all the principles are very helpful, I find the ‘Misconceptions’ area in each chapter very beneficial to my learning. In this part of the chapter, the author lays out many of the misconceptions and common mistakes students learning about the given strand will make, or be inclined to make. Familiarizing myself with these misconceptions when I am teaching math, may help me to alter my lessons to be more clear, hone in on these specific areas in order to avoid these mistakes altogether, or be more patient when students are struggling to understand certain areas.

This image was taken out of the Making Math Meaningful text, and highlights many students' misconceptions regarding decimals.

This image was taken from the Making Math Meaningful text and explores some common errors students have when learning fractions.

       

          Anyways, thanks for reading! I will be back with more updates from my math course in the upcoming weeks.

– Madeleine 

References:

Small, M. (2016) 3^rd  Edition. Making Math Meaningful to Canadian Students, K-8.3rd Edition, Toronto, Nelson.   

Catharine (2016). Confused student child. Miscw. Retrieved from https://www.miscw.com/confused-student-child-6297.html

Weekly Reflection 1

Hey everyone!

So, I have a few things I want to discuss in my first blog post about what I have experienced thus far along my math journey.

Math Refresher Course

I wanted to start off with discussing my experiences with the Elevate My Math refresher course. At first, I was very discouraged, as I was very overwhelmed by the length and level of the questions I was expected to answer. As mentioned before, I haven't had to use math in a very long time, and to my dismay, I had forgotten everything I had once learned. The questions were very challenging, and I found myself wondering and complaining about the whole experience – claiming things like, "I definitely did not learn this in grade 7 or 8" or, "I can't believe I don't know this, everyone else probably didn't even struggle at all." Right away I was discouraged, unconfident in my abilities and anxious about how I was going to ever finish it.


This is where I want to pause and reflect. 


Within 10 minutes of trying to figure out question number one, I was frustrated. It's this attitude that I want to ensure stays at a minimum within my classroom – ideally, not present at all; however, I know this is a challenging goal. The more I thought about others knowing what to do, the more I thought of myself as a failure; this mindset only pushed me further and further into despair. It wasn't until I cleared my head and realized, I could only do the best that I could do – I wasn't going to get kicked out of teachers' college because of it! Once I put on that mindset, I began to feel more relaxed and able – my confidence grew, and if I didn't know an answer, I shrugged and said, "oh well, that is what the refresher modules are for." 

So why could I not come into math with this attitude?

As a teacher, I hope to create a classroom that challenges why we feel anxious about math. I want to foster an environment where students see something that they don't know, and instead of fearing failure, they are excited about the failure and challenge that awaits. I want my students to realize that their growth as a not only a student, but an individual as well, will only come through failures and mistakes. Instead of fearing that their desk partner got the answer before them and they don't know, I want them to ask that student for help and not see it as weakness, rather, an opportunity for growth. I want to create relationships that are two-way streets; I hope my students will teach me methods of learning and ways to get there, just as I will for them. With that being said, I want to discuss the idea of a growth-mindset from our first class.

Growth Mindset

I have included a short video below to summarize what a growth mindset is to you guys who may not be familiar with it. While this plays an important role within our personal lives, I feel it fosters great meaning within the mathematic community as well. Instead of straying away from challenging opportunities due to the fear of failure (a fixed mindset), individuals with a growth mindset understand that failure may occur, but see it as an exciting way to learn. Individuals with a growth mindset believe that their "basic abilities can be achieved through dedication and hard work" and are not "simply fixed traits" (Growth Mindset). I recognize myself that this is an area of improvement I need to work on – especially in regards to math. How can I teach my students to have this mindset and not fear failure if I can't myself?

Oh, and by the way, when I started talking to others in my program about the math refresher course, we ALL were experiencing the same feelings. Here I was thinking that I was the only one struggling! This is an experience that I hope will guide my attitudes within the math classroom.




Sprouts (2016). Growth Mindset vs. Fixed Mindset. Retrieved from https://www.youtube.com/watch?v=KUWn_TJTrnU&t=29s

Ways of Getting There

Lastly, I want to discuss this attitude in regards to teaching and understanding math. In order to actually have this attitude, the ways we, as teachers, teach our lessons have to allow for new ways of thinking. Math is starting to stress less about the aspect of time constraints, especially in areas such as developing fact fluency. Research shows that it is not about the speed of answering 2x3, rather, it is about our process of how we got to the answer. We each make meaning differently, so why should we all be expected to get to an answer in the same methods? The text highlights the importance of allowing students to explore, make their own algorithms and teach these findings to other students. Giving students opportunities to explain their thought processes is much more important than arriving at the traditional way to go about an answer. 

Figure 1

Figure 1 from the text shows a student's work who justified her ways of figuring out a math problem; her explanation makes perfect sense, even if it is not a traditional approach.

Figure 1

Figure 2 from the text shows a traditional style of teaching and answering a question vs an alternative method that may better suit a student's understanding.

References:

Small, M. (2016) 3^rd  Edition. Making Math Meaningful to Canadian Students, K-8.3rd Edition, Toronto, Nelson.   

Sprouts (2016). Growth Mindset vs. Fixed Mindset. Retrieved from https://www.youtube.com/watch?v=KUWn_TJTrnU&t=29s

Growth Mindset (2013). The Glossary of Education. Retrieved from https://www.edglossary.org/growth-mindset/

Student frustrated with difficult math (2015). Clipart Guide. Retrieved form clipart guide.com 

Welcome!

Anderson, Mark (2018). Math Cartoon #7227. Retrieved from andertoons.com

Hi everyone! Welcome to my blog, where I will be talking all things math! Now if you're anything like me, even hearing the word math sends shivers up the spine – hopefully I can help ease this anxiety around math, as I get more comfortable with it myself. This is my first year in the Teacher Education program here at Brock, and I am so excited to get things going and start my adventures as a teacher candidate. I graduated from Western University with a degree in Studio Arts and English Language and Literature, so it's obvious I'm no mathematician. However, I am super excited to get my math brain back up and running.


Growing up I really enjoyed math – then I hit high school. All of a sudden letters were being thrown into equations, and I was just not about that. I remember struggling a lot with math in high school, going in for extra help constantly in grade 12 functions, and never wanting to put up my hand in class due to the constant fear I was going to be wrong. So hearing about this growth mindset gets me excited because I wish it were the mindset when I was growing up; instead, there was only one correct answer, and only one right way to get to it. I want to teach my students that failure is the best way to learn, and in math, there is going to be a lot of it. I also hope to gain a lot more confidence in math, so that not only will I be able to conduct a clear lesson, but answer any questions students ask me. I'm looking forward to the lesson planning and teaching of lessons, because I know the best way to learn is from one another. Hopefully, I will be able to learn from my classmates and teacher, as well teach them new ideas and ways of learning. 


Anyways, those are the hopes; I should probably remember how to long divide and properly apply BEDMAS before I start teaching children. 

Thanks for reading! I will be posting every other week, so stop back to see what interesting things I've learned and how my math journey unfolds!