Math Blog # 12: Forth and Fear No Darkness

I have learned so much in these short weeks. Amongst all the strategies and lessons and examples I realized that I had lost something that I came in with. I had lost my fear. Yes math still makes me nervous and yes, sometimes I have nightmares about being the one person in class who doesn’t understand a math concept and every laughs at me and then they all turn into pineapples. The point is, I never ever thought I could be excited to go to a math class- scratch that- I never thought I would voluntarily ever take another math class again, let alone teach one! And yet here I stand! (or more truthfully, here I sit because I am not talented or fit enough to type standing up). I still have so many questions and so much to learn but at least I know that I can continue from this point armed with the resources and knowledge I have been given by my professor and my peers. In the spirit of truthfulness and hope, I wanted to make a list of things I’ve learned and ways I still need to grow. Perhaps someone out there might match with at least one thing on my list. In the end, this is more for me than anything else. I wanted to literally spell out where I’ve come from and where I want to go. This blog is not the neatest, smartest or most consistent one out there, but it is truthful. Every week I missed, every mistake I made, every fear I have voiced. This is where I am at. I would say it’s not a terribly good place but maybe I’m on the road to somewhere better. I’m sorry this post is a bit rambly and rough. For an English major I fear I’m really not doing a very good job of summing up my feelings.

Well then. On to the list.

Things I have learned:

Ask questions.

This is for teacher and student. Sometimes you might be struggling to figure out what is in a student’s mind. Ask them. Get them to tell you how they are feeling. On the other side of the teacher friend of mine says that she always tells her students that if they don’t understand something to ask a question. Don’t say “I don’t get it” that doesn’t tell why you don’t understand, what you don’t understand, ask a question and we will figure out the answer together.

Repetition

Once you understand what the class is struggling with, go back or spend more time than you originally intended and address the problem. This doesn`t mean you stop for each little problem, but if you see an error that is becoming a pattern, or if after a test or assignment you see that the majority of students have made the same mistake, pay attention to that. It might mean there is a gap in your teaching, or a gap in their previous knowledge or the concept might be more difficult to grasp than you had supposed it would be.

Cultivate Your Resources

This means knowing your curriculum documents inside out, knowing where to find the best resources for specific subjects, or, if there isn`t much to be found in the usual places, pioneer your own lists. The more you learn, the more you explore, the wider your lens becomes. And if you feel that you have something, a strategy or resource, that you would like to share, you should go for it. If nothing else, you have gathered your thoughts and you have reflected. That is a worthy pursuit.   

Ways I Can Grow

Read and Re-Read my Textbooks

To be honest, I just completely forget some aspects of math because it’s just been so long since I was in contact with them. The Learning Presentations, my textbook and the math refresher course definitely helped me remember a lot of the concepts that were gathering dust in the corners of my mind, but I know I still need a BIG review. I guess it’s a good thing I have several siblings in elementary school who would be more than willing to let me peruse their textbooks.

Getting Organized

This really needs no explanation. I now firmly believe that to be a good teacher you HAVE to be organized. And with math in particular, you need to plan ahead and make sure the lessons and units flow well into each other and figure out what do to when you start to fall behind. This is by far my worst quality as a student, possibly one of my worst qualities as a human. Because I think my lack of organization keeps me from succeeding more than I would like to mention. 

Ask. For. Help.

No matter what you are doing, no matter what problem you are facing. People want to help. And in a school community, there is help everywhere. UTILIZE IT. 

There is so much more I could say, and maybe one day I will. But for now, it is time to move on. In the words of King Theoden: forth and fear no darkness. 

Until we meet again,

Bernadette 

Oral Communication Blog: The Debate on Debate

The Debate on Debate

Debate is a pretty scary word to students who already find it difficult just to put up their hand for a question. Unsurprisingly, educators and students have very strong opinions on debate and whether or not its presence should increase or decrease in the school system. Within the supporters of debate there are certain factions. Some believe it should only exist in the higher grades where students are less afraid of speaking and stating their opinions, others argue that debate is most beneficial if used throughout the grades, starting young.

In both cases, I think it depends on class environment. If you have a classroom environment where students feel that they can state their opinions respectfully without fear, there are strategies you can use to introduce constructive conversing and arguing to young students. This article is from a blog called Cult of Pedagogy written by a teacher who taught elementary for over a decade and now teachers Teachers Education courses. She lists several ways she got her students talking, and under each strategy she includes a list of variations to accommodate different learners and classrooms:

Classroom Discussion Strategies

Going off a discussion I had with my peers about their experiences with debate in the classroom, I have compiled a list of challenges and misconceptions that come with debate, and how they can be avoided.

1) The POP POP 

There is always one or two kids in the class who always have something to say about everything. They are the first to speak and their word is law. They might have very good, constructive ideas, or in the case of Magnitude from Community, (pictured below) have nothing of substance to say but speak with such confidence that no one ever challenges them. Often times with debates, the students are tempted to allow this student to speak for the whole team. An organized and structured debate format would help with this problem. Keeping the students used to the statement, response, statement response format, even introducing a rule where the same person can't speak more than once, or keeping the responses timed.

2. Point of View

An interesting tactic that I have seen used in the classroom is picking a topic that is relevant and of particular interest to the students (perhaps pitting various forms of social media against each other or two popular bands etc.) and then, before beginning the debate, send pieces of paper around and get everyone to write down which side they are on. Then gather up all of the papers and place everyone on the opposite side of the choice they made. The students may be reluctant at first, but I have seen this work quite well, once they get into the debate they want to win so badly that they start thinking of arguments to make that go completely against their own personal opinions. This is a great way to get students thinking about an issue from a different point of view.

3. The Screamer

Since mainstream media, especially recently, really hasn't been providing students with good examples of real debating, many students may believe that to debate means to state your opinion over and over until things get ugly. With younger students, you might get responses like "well that's stupid" or "if you think that then you're dumb" and the like. They resort to name calling and immaturity when faced with a point they can't or don't know how to refute. With older students it manifests itself in almost exactly the same way, speaking over each other, scoffing, etc. The insults are just often more intellectual. Though not always. But debating is equal parts listening and speaking. Students should be taught from a young age how to respond respectfully to someone they disagree with, and introducing debate can be a great way for students to learn that. You could start with a list of phrases to use when you disagree: "I respect your argument, but have you thought of this..." or even highlighting where you agree before going into how you differ: "I agree that... but I feel that..." and then have the student go from there.

At the end of the day, I'm not sure how well debate would work in every classroom. The teacher needs to know their students and gauge whether they would benefit from a formal debate structure or if they would be better suited to a debate-like game like the ones listed in the Cult of Pedagogy article above. I do, however, truly believe that there needs to be more of an emphasis on Oral Communication in the classroom than what exists now. Maybe this means debate, maybe this means more presentations, whatever shape it takes, students need more opportunities to exercise their communication skills from a young age.

Math Blog #8 : Making Mention of Meaningful Measurement Materials

This week I was introduced to an AWESOME resource for teaching Measurement! Measurement Grades Four to Six. Wowie wow. I guess one of the great things about being a teacher candidate is that everything is new and exciting. Whenever I am introduced to a new resource or concept I immediately start thinking about how I could use it and with math in particular (in case my past blogs have not made this evident) I need all the help I can get.

My biggest worry is actually not how to get the students to understand math, its figuring it out on my own and feeling confident enough to teach it correctly. So when I am given a resource like Measurement Grades Four to Six I get very excited indeed. This document breaks Measurement down so well. I love the way this whole document is structured. The language is engaging and accessible and the learning goals are very clearly laid out. But my favourite part of this document is the section on Learning Activities. Each grade comes with two or three fun activities that help students build a conceptual understanding of measurement and apply it to real life. I am ten thousand percent going to use this document in the future. And I mean the very near future. I have already been creating questions for my placement class and I am definitely going to try to make use of this document!

My placement class is currently in the measurement unit and as I marked their math worksheets I could see patterns forming around what the students got right and what they got wrong. Many students could easily remember how to calculate the perimeter of any shape but would often make the mistake making their answer squared. My associate teacher said that many of the students didn't learn why area was expressed as a number squared, they just knew that they had to do it. And so some students were making their perimeters squared as well. So what my teacher does is what this resource strives to do, teach the students a conceptual understanding of the concepts they are learning so they can make logical decisions.

Writing Blog: Helping Students Get Started

As a future language teacher I am constantly looking for resources that will help me shape my teaching style and be the most effective educator possible and many of the articles, discussions, or people that I meet say that writing is an area of the curriculum that poses many problems.The Ontario Language Curriculum  states that students need to become "disciplined thinkers" to be able to write effectively and this comes from being given many opportunities to write (2006). The problem is not that students are not getting these opportunities but rather that they often don't get used to their fullest potential. I have discussed this theory with my fellow teacher candidates and several have said they notice that students struggle to begin writing, but once they start the process is easier. I read a great article on a method to get students thinking before they write here. But I wanted to share a personal example of a method that could help create a balance between overwhelming amounts of choice or lack of freedom in classroom writing.

Last week in my Cognition and the Exceptional Learner course I was placed in a group and we were asked to find ways to use a certain resource or method in teaching language or math. We were given Cubing, a method where the teacher creates a cube and writes a different option for completing a task on each side. We decided to use it to help prompt students to exercise different types of writing. I snapped a picture as we were planning:

As the picture shows, this cube could be easily modified to fit each groups needs and abilities. One cube would have different perspectives to choose from (first person, third person, a dog, a tree) or sentence prompts that can go anywhere in the story, beginning, middle or end. And we would allow each student to roll the cube three times and choose their topic from the three so they wouldn't be completely limited but still have a much smaller amount of choice to pick through.

 I dearly wish I had come across more "games" like this one in my formative years. In my early years of elementary school I spent a couple parts of my day in the "resource" room where I received extra help in writing and math. Anyone who has had this experience will probably agree that its never the place you want to be. Though the resource teachers were lovely people who tried their best to help me, I felt there was always a disconnect between what I learned at resource and what I learned in the classroom. Like my resource teachers had been given incorrect or incomplete information on me. The resource work was far too easy and rarely connected to the concepts I was working on in class and I would be missing my real classwork and feel further and further behind. This was my personal experience and I can acknowledge now that my bouts of self-pity and laziness would keep me from achieving my goal more often than the resource visits, but I still wonder what kind of writer I would be I had had the chance to stay in the class and still learn at the pace that I needed.


Ministry of Education. (2006). Ontario Curriculum Grades 1-8. Ontario, Queen’s Printer. Web. Retrieved from http://www.edu.gov.on.ca/eng/curriculum/elementary/language18currb.pdf

Math Blog #7: Strategies for ESL Students

Math already comes with a unique set of challenges due to its negative stigma, its mix of number and word problems, and the countless equations (and exceptions) students are expected to know (I'm hiding my English bias so well aren't I?). Then take all of that and imagine it was in a language you barely know. This is the reality for ELL/ESL students, and keeping up with math might seem daunting for both student and teacher, but educators everywhere have been sharing their strategies for teaching math to ESL students in the hopes of making the process easier for student and teacher alike. I'd like to share some of those strategies here:

My math professor Rebecca Bunz has had experience working with ESL students and she shared these strategies with our class:

- Provide concrete examples, models, etc
- Simplify the language not the ideas
- Introduce new vocabulary with contextual support
- Have students keep a personal dictionary; can write definitions in both languages
- Non-verbal cues such as gestures and body language
- Give extra processing time
- Spend extra time to ensure they understand the problem

- Have them rewrite problems in their own words

On my search for ELL/ESL strategies I read an article called "Math Instruction for English Language Learners" on a website called Colorin Colorado, an educational website that provides free research-based information, activities, and advice to parents, and schools on ELL learning. The author, ELL specialist Kristina Robertson writes in the introduction to the article that many people make the assumption that math is a universal language and that many ELL students will excel at it. But she points out that this is not necessarily true, especially since many of these students may lack a prior knowledge of math terms we consider common. And so she goes on to give her advice for math teachers with ELL students and there are a few points I'd like to share:

1) The Importance of Teaching Academic Vocabulary

Teaching the difference between the definition of a word and the mathematical definition of the word. As shown here, expressions like "find X" can be very confusing because it could be taken literally.

- make sure to explain that words can have multiple names

- encourage students to offer bilingual support to each other

Robertson, K. Equation. [Online Image]. 
Retrieved from http://www.colorincolorado.org/article/

2) The Importance of Reading and Understanding Written Questions

She included a good quote from Reading and Understanding Written Math Problems that sums up the problem quite well: "Word problems in mathematics often pose a challenge because they require that students read and comprehend the text of the problem, identify the question that needs to be answered, and finally create and solve a numerical equation — ELLs who have had formal education in their home countries generally do not have mathematical difficulties; hence, their struggles begin when they encounter word problems in a second language that they have not yet mastered" (Bernardo, 2005).

Word problems are generally the most common issue for ELL students because of the language barrier. She suggests explicit instruction of key vocabulary, daily practice of problem solving, repeated readings of the word problem together as a class, and hands-on activities such as movement, experiments, or drawing to help students comprehend the problem (Robertson). 

Reading Strand Blog: Effective Reading

Language literacy is so often taken for granted in our society. When I completed my undergraduate degree, I received countless “polite” inquiries on what I hoped to accomplish with a degree as useless as English. To answer them I first politely corrected their pronunciation of the word useful, and then attempted to express the immeasurable value of spending four years building reading, writing, and communicating skills. Being able to communicate your thoughts and ideas clearly, succinctly, and persuasively in any format is an enormous asset to any job. Knowing how to summarize information, how to process it and retain the important details, and then knowing how to identify which details those are, these are all skills that go beyond a classroom or an office, they follow us into our everyday lives.

Reading is one of these skills we as a society so often take for granted. Someone might say they aren't very good at writing and it wouldn't be considered particularly strange because that statement would most likely be understood to mean the person might not be an interesting writer, or a persuasive or creative writer. However, if a grown adult told you they couldn’t read very well it would sound slightly more concerning because reading as a concept is often considered at its most basic level. When a person thinks of reading it is often of the surface level, just the ability to look at words and understand what they mean on their own and how they work together to form a sentence, but being able to read does not make you a good reader. The Grades 1- 8 Ontario Curriculum states that elementary students need to learn to become “effective readers”, to go beyond just understanding the ideas in a text and to start making connections between past knowledge and applying the new information to new contexts (2006).

IMAGE: TUMBLR SHHHHHHIMLISTENINGTOREASON. Retrieved from mashable.com.

As a teacher you need to be able to teach students to think critically about everything they read because, as I discussed in my previous literacy blog post, the world is throwing information at you every second of the day and you need to be able to interpret what is important, what is real and how it applies to you and the world around you. And this process may look different for each student. Some may have to write their ideas down or need to work in groups to hear other perspectives which may spark their own ideas (a great resources for this is Padlet.) Some student may need more time to process these ideas and though it may it seem concerning when a student is reading slower than most of the class, it is our job as teachers to understand that not every student has the same process. I have always been a slow reader and my father who is a professor is also a slow reader and it is not because we lack the skills to recognize the words, as my elementary school teachers mistakenly believed, but because we read and reread sections of the text in our head that we find interesting and important. It is also important to remember that text takes many forms: prose, poems, journals, plays, films, radio etc. and students need to be exposed to these different types of text in the classroom so they understand how to adapt their effective reading skills to any text they may come across in the future. 

Blog #1: It Begins!

Hello! I'm Bernadette and I am a Teacher Candidate at Brock University! I will be using (and have already used) this blog as a tool for my various courses!

This is me! Blissfully unaware of the
math that awaited me 

I hope this course will teach me how to reduce the negative attitude towards math that I and most students I have encountered share. Math is a terrifying thing to many students and I think the reason for that is that it has a reputation of being something extremely difficult with very little that is applicable to the real world. I hope this course will reduce my fear of failing as a math teacher and allow me to grow an appreciation for how important math is and how it can be made more accessible!

Math Blog #6: Coincidence? I Think Not!

So, on my last blog post I wrote about my fear of not being able to explain difficult concepts to students and then in class the next day we talked about ways students can misunderstand statements, processes, or the "why" behind a math problem, and the resources we as educators can use to combat these problems! Coincidence? It is a simple truth that one of the greatest dangers of teaching math is the temptation to only explain a concept the way you have understood it and not thinking about other ways to approach problems that might benefit students with particular learning styles. Teachers may feel constrained by time, worrying that working too long on one concept will not leave enough time for another, and have to strive to find a balance between how they would like to teach and how their curriculum expectations will allow them to teach. I believe that in this increasingly digital world there will be a greater need for the kind of problem solving skills you learn in math and that the attitude that math is boring and useless is shifting each year towards a more positive practical one. But this shift will only continue if teachers and teachers to be are dedicated to creating that change.

Smith, Dustin. (30th August 2012). Math Quote. [Online Image].
Retrieved from propensityforcuriousity.com.

I came across a blog last week written by Brie Finegold, co-organizer of Women Advancing Arizona Mathematics, that had some interesting things to say on this topic. It was called My Top Ten Issues in Mathematics Education and while I didn't agree with everything on the list and some aspects referred more to the American education system, there are two parts that I think are worth mentioning.

The first is that "Discovering and uncovering content should take precedence over covering and recovering content." (Finegold, 2014). While many educators make the case that students benefit most from building a conceptual understanding of the math, I agree with Ms. Finegold that talking and talking while students are trying to wrap their heads around a concept is not very helpful. She gives a link to a list of suggestions for becoming invisible for teachers who find it difficult not to phrase and rephrase things to fill the silence. 

The second is that "Mathematics Educators deserve opportunities to further their own content knowledge for teaching." (Finegold, 2014). I think this also applies to teachers in general, but math teachers are rarely given the opportunities to try new, well researched, teaching methods that they believe may make a needed improvement to the education system. Often times a person will teach as they were taught "regardless of whether it was truly effective" (Finegold, 2014). 

I think another issue that should be on that list is the use of phrases like "obviously" or "this concept is simple" or "how can you still not understand that?" that are obviously very damaging to a students confidence and perpetuates the negative attitude in mathematics. 

Lastly, I would like to say that all three of the presentations this week were prime examples of the kind of creative examples a teacher can use to help explain a concept. All of the presentations were clearly very well researched and each activity brought a unique insight on Rates, Ratios and Proportions.

Math Blog #5: Fighting Helplessness

Just as I sat down to write out my my post for this week, I realized that most of my posts stem from interactions I have had with my siblings and not with students in a classroom and I should explain that I am not yet in my placement and so the only way I can observe math instruction in action is through the tales my siblings tell of their teachers and the homework they bring back. And so, today when my little sister came home from school and asked me to help her with her math homework I was very eager to assist, but I found that I was not as helpful as I would have hoped I would be. She was doing some sort of patterning work that I hadn't come across before and she didn't have her textbook at home so I couldn't look it up. And I felt this painfully familiar feeling of helplessness creep up my spine. A feeling that I have been having less and less since I started this math course, but still very much present.

Last Friday's class for instance, when we talked about explaining Integers and we were introduced to the soup analogy which is shown in this video:



I loved the idea of this method, it breaks down the concept and gives very helpful visuals. I only had an issue with explaining the Zero Pairs. For example let's say that you had to solve 3-6 = -3. In our exercise in class the professor added three hot cubes (positive) and then added six cold (negative) and six hot to represent the zero pairs you would be pulling the answer form.

Here is where the feeling began to creep back.

I understood how that process worked but I also immediately identified a way that a student could very easily be turned around. With every example that we did we were told that hot and cold cancel each other out. But in this instance that doesn't happen. You take what you need from the negatives and then what is left cancels itself out. But I could easily see a student cancelling out the zero pairs and leaving positive three as their answer, And I was worried that if I student did make this mistake, that I wouldn't be able to help them understand the concept. I think fighting this feeling of self doubt and fear will be my biggest challenge during this course and I fully intend to face it head on. A good teacher is someone who is constantly self improving and self reflecting that is what I strive to embody. I took to the Internet and I found this video above which elaborated on zero pairs in a different way than my professor. So now I will be equipped with two ways of approaching this concept if a student does have a question, and I already sense the helplessness fading.

Media Literacy: Developing a Critical Eye

   Media, like reading, writing and oral communication, is so deeply integrated into our daily lives that we don't realize how often we actually come into contact with it. Films, television, advertisements, images, music, it’s present in nearly everything we do. When young children start to become exposed to these different forms of media in larger and larger amounts, parents can’t be monitoring them all the time and so it is the educator’s job to teach young students to think critically about the media that surrounds us and analyze its messages. 

   In this blog, I want to discuss ads and how they can be an extremely useful tool for improving a student’s media literacy. I’ve been on an SNL kick recently and so I’ve been on YouTube more than usual looking up short sketches to watch during study breaks. I clicked on a promising looking sketch featuring Emma Stone in an insanely huge wig when an ad started playing. It was one of those super annoying thirty second ones that you can’t skip. And as I was forced to listen to middle aged women tell me why I needed to buy a blender I started thinking about my younger siblings who spend a lot of their time on YouTube, and I wondered if they were getting the same ads that I was or if for some reason my watch history had somehow informed YouTube that I might likely be a middle aged woman in need of a blender. It’s easy to believe that we are all impervious to ads and that our brains just ignore the messages they are subtlety or not so subtlety trying to deliver, but the messages conveyed by the media impact us all and helping young students recognize this fact is a great way to examine just how media may have shaped the way they perceive the world. Or their understanding of how their gender, culture or ethnicity is often perceived. 
   TEDxTalks is currently running a spotlight on bold young women who challenge stereotypes and sexist or unrealistic portrayal of females in the media, education and society. Many of these young girls formed their foundational knowledge on what they were capable of from their parents, educators and peers, which serves as a reminder that we all have a part to play in fostering environments that break down these barriers. 

    A good exercise to open up this kind of reflective thinking would be to give each student in your class a brand or company whose target audience is the age group of your class, or just young people in general, and ask them look up and choose a couple ads for this brand and write down similarities in the marketing, the people they see featured in the ads, the people they don’t see, how they are portrayed, how it made them feel, and have them present their findings in groups and discuss the messages they thought the companies were trying to convey. As the Ontario Language Curriculum states, it is so important for students get in the habit of questioning media or any form, breaking it down and learning to separate myth from fact. Many activities of a similar nature are being used in classrooms all over the world so students can be equipped with the tools to approach any form of media with a critical eye. 

Ministry of Education. (2006). Ontario Curriculum Grades 1-8. Ontario, Queen’s Printer. Web. Retrieved from http://www.edu.gov.on.ca/eng/curriculum/elementary/language18currb.pdf

Math Blog #3: Resources!

I decided to make a blog full of resources I have found useful for teaching and understanding math! Most are suggestions from my professor but some I found on my own. I'm planning to continue building upon it as I learn!

Rainbowgram. (14 January, 2015). Penguin. [online image].
Retrieved from rainbowgram.wordpress.com

Khan Academy

Khan Academy was my savior during summer math course. The videos are easy to understand and if you watch them on the website they come as part of a step by step process to understanding a concept. I have used it countless times since, most recently to help by little sister with percentages!

Inspiring Math Videos

Whether its success stories, new perspectives or simply a deconstruction of a math concept in an interesting way, these videos show the importance of math in our lives:

Brain Crossing

Math Class Needs a Makeover

EduGains

I have used this resources several times for my literacy course and only recently discovered how helpful it can be for math! It provides educational articles, examples of lessons, links to other resources and important mathematics documents.

Math Interactives

This interactive mathematics resource allows students to explore the pictorial, symbolic, and concrete representations of fractions.There are examples of activities you can print, a smoothie game you can play and other fun resources! I also like that it is a Canadian site. 

Battleship Jeopardy

AMAZING game created by my professor Rebecca Bunz! It fuses Battleship and Jeopardy to create a super fun integer game!

Math Blog #2: Lord Voldemort

Math Blog #2 

Even if I loved math it would be impossible to say that it doesn't have a negative stigma, and since I definitely do not love math, I can truthfully say that I have experienced the negative environment it creates first hand. I like to compare math to Lord Voldemort. The name itself strikes fear into the hearts of all except those who can see it for what it really is: a negative association that you can overcome.

James, Bernadette. Meme Generator. (September 21st, 2016).
[online image]. Retrieved from https://memegenerator.net/. 

I think a big problem I had with math as a kid was mastering something and then feeling like I never used it again. When we read over the Math Curriculum Grades 1-8 it was very comforting for me to see the continuity of the mathematical strands and expectations. As I become more familiar with the concepts that weave their way through the grades, it will be easier for me to make the connections clearer to the students. For example, common multiples (the concept I am working on for my Learning Activity Presentation) start with whole numbers, then can be used to find the lowest common denominator fractions. It is a good example of proportional reasoning, being able to compare quantities through multiplicative thinking, which in turn is part of the Number Sense and Numeration strand of the Math Curriculum. When you take the time to think about it, it all connects, or at least it should connect, rather neatly. But sometimes it doesn't, as I noticed in my readings.

In Chapters 10 and 11 there are little teaching tips and sections for problems that children often have with the concepts covered that you wouldn't think of. For example, one blurb read that children struggle with "teen" numbers, and sometimes if they hear 17 they will write it as 71.  It's been so long since something so simple could have been difficult and I think its challenging for teachers to get back in that head space. I believe teachers who experienced these kind of difficulties will be able to be more patient and understanding, and in doing so, contribute to the creation of a positive math environment.

Ministry of Education. (2005). Ontario Curriculum Grades 1-8. Ontario, Queen’s Printer. Web. Retrieved from http://www.edu.gov.on.ca/eng/curriculum/elementary/math18curr.pdf