Week 4 - Report & Reflection

Hello fellow math people! I hope you're enjoying your week finding math in your every day life. For me, this week was rich! Specifically, I learned what mathematically rich tasks are and why they are important to do in your math classroom. 

Rich tasks - huh? 
Please do not confuse these with tasks that will make you rich. That's not what I'm explaining here. If I knew those tasks, I'd be working on them as you read.

According to Steve Hewson, mathematically rich tasks are activities that offer different opportunities to meet diverse learner needs at the learner's own pace. Hewson also explains that these rich tasks allow students to extend or apply knowledge in a new context, work using different methods, make connections between the real world and math as well as between different areas of math, and develop problem-solving skills. This technique also takes the pressure off of memorizing number facts and emphasizes the process. Students embark on problems that may not have a clear path to a solution, which allows students to explore a variety of possible skills and adjust or defend their thinking as they go.

That sounds like a lot! So why do it? 
With a task so loaded it may seem that the benefits get lost in the details, but that is exactly where they shine the brightest! Rich tasks allow ALL learners to find something challenging for their level. This provides students with the opportunity to question and develop their understanding of mathematical ideas and gain confidence in applying their knowledge in a variety of contexts as Hewson said. 

In this video, Diane Briar explains the importance of having students use mathematically rich tasks.

Rich tasks foster a growth mindset in math class because there are so many more possibilities for students to arrive at a solution and students know that. They know that intelligence can be gained by working through any one of these possible paths. With rich tasks, it is also easier to understand that mistakes and risks are necessary because they all point us in the right direction. Applicable to many levels, students feel safer to take those risks, make errors and most importantly, learn from them.

My experience of working on a rich task
This week in class we had a few opportunities to try out some rich tasks. At first I found it frustrating because there were so many questions I had about the library and swimming pool question. For me, there were too many variables and openness for me to get started. In my opinion, students may get lost in asking all of these questions if they are not confident or unsure of how to get started. However, once we talked about all those variables as a class it was a bit clearer and I felt better prepared to get started.

My attempt at writing a rich task
In class we were given the following data set and asked to come up with a rich task to teach mean, median and mode.

70 98 83 87 65 70 88 75 71 77 90

The numbers above are the points scored by the Raptors in their pre-season games. Make a prediction about how well the Raptors will do this season. 

Tipton, M. (2013, June 28). "2013 Toronto Raptors 3" [Online Image]
Retrieved from https://bit.ly/2DAGMxU

Here are a few ideas what students could do:

· Find the mean to determine the average number of points the Raptors scored and make a prediction based on that number (What is a good average score to do well?).

· Find the median to determine the average number of points the Raptors scored and make a prediction based on that number (Is this an accurate representation of how well they will do?).

· Find the mode of the number of points the Raptors and make a prediction based on that number (Is this an accurate representation of how well they will do?).

· Order the numbers from smallest to largest (assuming that this the progression of points scored in consecutive games) and use that as a point to argue the Raptors scored more as they played farther into the pre-season.

· With Lebron now in the western conference and the data set, students can make a prediction.

· Look to statistics from the previous pre-season and compare them with this data set to make a prediction.

Please feel free to let me know how I can improve this rich task!


Pun of the Post
What do you call friends who love math? Alge-bros!

Week 3 - Report & Reflection

Thanks for visiting my blog and checking out what I have to say about mathematics. I hope you find this post especially useful, because I know when I learned about open and parallel tasks this week, mathematics opened up for me.

Open & Parallel Tasks - What's That?

This is the exact question I asked myself when I heard these terms.

Open tasks allow for a broad range of responses at many levels. This entices students from all levels of mathematics to participate, according to Marian Small's presentation DifferentiatingInstruction in Math: It's Not as Hard as You Think. Open tasks broaden everyone's learning and makes students feel like their contributions make a difference, which in turn boosts confidence.

My instructor also provided some strategies to create open tasks.

McEachren, P. (2018, Sept. 17). Strategies and Examples for Open Tasks. [Chart].

Parallel tasks are activities given to children where they have the choice between two tasks that "focus on the same key concept, yet address students at different levels of mathematical sophistication" as The Literacy and Numeracy Secretariat put it in their monograph Differentiating Mathematics Instruction. However, they are designed so that a whole range of students can participate in a discussion about them, regardless of the task chosen.

Both open and parallel tasks should encourage students to think mathematically and not just rely on procedures and algorithms.

Importance of Questions

One of the important things to do as a teacher while students are working or about to begin working on an open or parallel task is questions. There are two types: generic and scaffolded. Generic questions can be asked of both tasks, and aim to address the big idea at hand. Scaffolded questions are more specific to which task the student chose and acts as a support to get the student started. Scaffolded questions allow the teacher to "be less helpful" in that it gradually releases responsibility from the teacher to the student in finding the answer on his/her own and working independently successfully.

Why Differentiate in Mathematics?

This answer to this question may seem obvious to you, but I'm writing this part of the post for those who believe math is understood through memorization of algorithms. This is nonsense! Everyone learns differently, simply because we're all different beings.

A lot of the skills learned in mathematics, such as reflecting, communicating and reasoning are valuable skills used in everyday life. In order for all our students to become proficient in these skills, they need to learn them in contexts that make sense to them and are meaningful to them. Differentiating instructions provides those contexts for optimal learning for ALL students.

Resources for Teachers

The Ontario Teachers' Federation has a resource bank full of mathematics lesson plans, which incorporate open and parallel tasks at various levels.

This presentation Differentiating Instruction in Mathematics: It's Not as Hard as You Think by Marian Small has a lot of examples of open and parallel tasks.

Pun of the Post:

If Gotham was a mathematical city, where would all the bad guys go? To the prism!
(Special thanks to one of my students for sharing this joke with me)

Week 2 Report & Reflection

Welcome back to Math with Markers, where math is more than pencil and paper!

This week, I read Paying Attention to Spatial Reasoning: A Support Document to Paying Attention toMathematics produced by the Ontario Ministry of Education. When I first looked at the article I asked myself A LOT of questions like: What is spatial reasoning? Do I have it? Can I get it, if I don't have it? Where can I get it? Is it available on Amazon Prime? Okay, maybe that last question didn't really cross my mind, but it is a question that could have guided my search for spatial reasoning, regardless of the answer.

What is spatial reasoning?

I found that spatial reasoning, or thinking, refers to the ability to think, mentally or physically, about the location and movement of objects in three dimensions and to draw conclusions about these objects.

It integrates three components:

1. Concepts of space (understanding relationships within and between spatial structures);

2. Tools of representation (using a wide variety of possible representations, such as drawings or computer models); and

3. Processes of reasoning (means to communicate about these spatial structures).

Still not sure what it is?

Think about geometry in math class determining the volume of a soup can. 

Or being aware of your body's position in phys. ed. class so you don't hit anyone while you're stretching.

Or driving your car on the highway and having to stay in your lane and a safe distance behind the car in front of you.

Ministry of Education (2014). "Spatial reasoning can involve"
[Screenshot]. Retrieved from MOE website

Why do we need it?

Spatial reasoning is used in everyday life, as demonstrated by the examples above. We need good spatial reasoning skills to further the advancements in sciences, technologies, engineering and mathematics (STEM). But we also need it in many of the arts, such as architecture, graphic design and geography. I didn't think I had good spatial reasoning skills until I figured out that essentially, it helps us navigate our way through this 3-D world. So if you don't think you are a "math person" (although we all are - that's a topic for a different post) or an artsy person, you still need strong spatial reasoning skills to be successful in all aspects of life.  

How can we help students develop spatial reasoning skills?

- Give students physical manipulatives to build those spatial reasoning skills.

- Encourage students to draw out the problem at hand or use pictures to explain their thinking.

- Encourage students to represent their thinking with diagrams, maps and graphs.

- Use spatial vocabulary, such as words related to distance, location, rotations, translations, transformation, and direction, such as north, south, east, west, left, right, near, far, close, below, above, middle, over, under, etc.

I have decided to take it upon as a math goal for myself as a math teacher: to explicitly teach spatial reasoning skills so that whatever those students grow up to be, they'll have the skills to be successful!

Pun of the Post:

What do you call a parrot that is dead? A poly-gon!

(Special thanks to one of my students for sharing this joke with me)

Week 1 Report & Reflection

Thank you for coming back to my blog! I know it has been a while since I have posted. I took some time off this summer to enjoy the finer things in life, like family, friends and of course, math!

With the start of the new school year comes new classes and new challenges. I am in a second year mathematics course and could not be happier with the challenge my instructor has set out for us as teachers: having a good math mindset. What does that mean? It means having a good attitude and an open-mind when it comes to teaching and learning math. It means persevering through the confusion and frustration that many of us face when doing math.

Scaffolded Math and Science (2018). We Are All Math People
Growth Mindset [Poster]. Retrieved from TeachersPayTeachers.

How do you get this good attitude and keep an open-mind? You create it for yourself! Recent research shows that the level of which you do well in math comes from the beliefs you have and the experiences you hold about it. (Watch Stanford Math Myths and the Brain 2 for more info.) This emphasizes the importance of fostering a growth mindset in ourselves and in our students.

I'm sure I have mentioned before that I teach math, and in doing so, I have seen students with and without this math mindset. The students without it struggle because they lack the confidence to try in fear of failure. Interestingly, they do not just struggle with math. They struggle with other subjects too for fear of failure and looking dumb in front of their peers.

On the other hand, students with a math mindset overcome more than they can ever imagine! As I am writing this, there is one student in particular of whom I am thinking and he was performing well below his grade level in math. However, he never gave up, despite his dislike for math and the negative emotions he associated with math. He was always willing to try new problems and determined to get them right, even when it seemed he had tried every other wrong way imaginable. I am proud to say that he is now working beyond his grade level, and in some cases two grades, thanks to his math mindset and perseverance.

Seeing the effort and willingness this student has demonstrated allowed me to see that all students can perform well in math. There is no "math gene" in any of our DNA. We are all capable of doing math and we are all capable of doing math well.

In the words of the infamous Jimmy MacElroy of Blades of Glory, "If you can dream it, you can do it!"

Pun of the Post:

I'll do algebra, I'll do trigonometry, and I'll even do statistics. But graphing is where I draw the line!