Chapter 7: Context as a Setting

con·text

/ˈkäntekst/

noun

1. the circumstances that form the setting for an event, statement, or idea, and in terms of which it can be fully understood and assessed.

   "the decision was taken within the context of planned cuts in spending"

Context...the thing I am trying to give to you right now...is most simply put, "the setting". I really like that term, as it refers to some sort of "place" in our minds. The context of a statement or idea should paint a picture of where that information belongs.

I could say to you, "I'm so sorry I was late today, I blame it on my truck". There are a million different ways that could be interpreted. Did my truck not start this morning? Did I have a flat tire? Did I get stuck in the snow? Do I just love driving it so much that I took the long way?

Without context, you would never really know why I was late (however, if you've seen my truck, you might be able to guess that it probably did not start).

Okay, so context is important in day-to-day conversation... but it doesn't stop there. This week in class, we discussed the use of context in the mathematics classroom. More specifically, we investigated the use of context to "hook" students into a situation which required them to use mathematics to solve/understand.

 For instance, we looked at Loblaw's initiative of diverting 1 billion plastic shopping bags from landfills by the end of 2009. Diverting 1 billion plastic shopping bags from our landfills seems like a really great initiative, but is it really that impressive? What does 1 billion shopping bags look like? That is such a large value, that it is difficult to really conceive. If you haven't already, try it. Think, what does 1 billion shopping bags look like? The size of a bus? A room? A house? A warehouse?

Its a very thought provoking idea, and as it did for me, this concept "hooks" students into wanting more. They want to figure it out...they want to figure out what this value "looks like". From here, there are an infinite amount of ways the lesson could progress, but that fact is, the context of the problem was so thought provoking that students wanted to find the answer.

I think the really cool thing about mathematics, it that it is everywhere around us. Whether we consciously choose to or not, we use math every single day to solve the many problems we encounter. This innate property of the subject can, and should, be harnessed by every mathematics educator to get students emotionally invested in the curriculum.

As you;re reading this, you may be thinking "Wow, this is really great, but it seems like such a daunting task!". I couldn't disagree with you more. Providing context doesn't always have to be an elaborate story or situation. If the problem captures their imagination or resonates with their life, they will be much more likely to become involved with the problem. For instance, when introducing the Pythagorean theorem, don't just use a triangle. Use something that is current/relevant to their lives. A lesson that a colleague and myself developed earlier this year used the concept of "Pokemon Go" to demonstrate the use of the Pythagorean theorem (see below).

This problem isn't any different than a regular old "given two sides of a right angled triangle, solve for the hypotenuse using the Pythagorean theorem" question, however providing context which is relatable to your students will generate very different results (and only took an extra 2 minutes to make).

As mathematics educators, we face the challenge of changing the perception that mathematics is "boring" and "too hard". I think that providing context to the problems we pose in the classroom is a great start.

Thanks for reading!

Comment below if you have any questions, also please consider the following questions to debrief my post:

-is there such thing as bad context?
-should we provide context for every problem we pose?
-are there dangers of providing "too much context" in a given problem?

Chapter 7: Context as a Setting

con·text

/ˈkäntekst/

noun

1. the circumstances that form the setting for an event, statement, or idea, and in terms of which it can be fully understood and assessed.

   "the decision was taken within the context of planned cuts in spending"

Context...the thing I am trying to give to you right now...is most simply put, "the setting". I really like that term, as it refers to some sort of "place" in our minds. The context of a statement or idea should paint a picture of where that information belongs.

I could say to you, "I'm so sorry I was late today, I blame it on my truck". There are a million different ways that could be interpreted. Did my truck not start this morning? Did I have a flat tire? Did I get stuck in the snow? Do I just love driving it so much that I took the long way?

Without context, you would never really know why I was late (but if you've seen my truck, you might be able to guess that it probably did not start).

Okay, so context is important in day-to-day conversation... but it doesn't stop there. This week in class, we discussed the use of context in the mathematics classroom. More specifically, we investigated the use of context to "hook" students into a situation which required them to use mathematics to solve/understand.

 For instance, we looked at Loblaw's initiative of diverting 1 billion plastic shopping bags from landfills by the end of 2009. Diverting 1 billion plastic shopping bags from our landfills seems like a really great initiative, but is it really that impressive? What does 1 billion shopping bags look like? That is such a large value, that it is difficult to really conceive. If you haven't already, try it. Think, what does 1 billion shopping bags look like? The size of a bus? A room? A house? A warehouse?

Its a very thought provoking idea, and as it did for me, this concept "hooks" students into wanting more. They want to figure it out...they want to figure out what this value "looks like". From here, there are an infinite amount of ways the lesson could progress, but that fact is, the context of the problem was so thought provoking that students wanted to find the answer.

I think the really cool thing about mathematics, it that it is everywhere around us. Whether we consciously choose to or not, we use math every single day to solve the many problems we encounter. This innate property of the subject can, and should, be harnessed by every mathematics educator to get students emotionally invested in the curriculum.

As you;re reading this, you may be thinking "Wow, this is really great, but it seems like such a daunting task!". I couldn't disagree with you more. Providing context doesn't always have to be an elaborate story or situation. If the problem captures their imagination or resonates with their life, they will be much more likely to become involved with the problem. For instance, when introducing the Pythagorean theorem, don't just use a triangle. Use something that is current/relevant to their lives. A lesson that a colleague and myself developed earlier this year used the concept of "Pokemon Go" to demonstrate the use of the Pythagorean theorem (see below).

This problem isn't any different than a regular old "given two sides of a right angled triangle, solve for the hypotenuse using the Pythagorean theorem" question, however providing context which is relatable to your students will generate very different results (and only took an extra 2 minutes to make).

As mathematics educators, we face the challenge of changing the perception that mathematics is "boring" and "too hard". I think that providing context to the problems we pose in the classroom is a great start.

Thanks for reading!

Comment below if you have any questions, also please consider the following questions to debrief my post:

-is there such thing as bad context?
-should we provide context for every problem we pose?
-are there dangers of providing "too much context" in a given problem?

Chapter 6: Engaging with Technology

As you're reading this, pause for a moment to observe your surroundings. Aside from the computer you are using to read this blog, how many technological devices are you near? My answer was 5, and it is my guess that yours is somewhere in that neighborhood as well. Technology is everywhere in our day-to-day lives, assisting us with countless tasks; why can't the same be said for learners in the classroom?

This week in class we discussed the use of technology in the mathematics classroom. Some of the topics which I found to be particularly thought provoking included:

-What are some appropriate uses of technology in the mathematics classroom?
-How can we evaluate a technological device/platform?
-When should we allow students to use calculators in the classroom?
-Does the curriculum explicitly demand students to use technology in the mathematics classroom?
-How can we use technology to align practice to curriculum?
-What are some pros/cons of technology?

Feel free to discuss your opinion of these topics in the comments below.

How can we evaluate a technological device/platform?

The SAMR model is tool used by educators to help evaluate the use of technology in the classroom. SAMR is an acronym for substitution, augmentation, modification, and redefinition. Using these four criteria points, teachers can evaluate whether or not the use of a given technology is appropriate for its intended use. For instance, if a teacher asked students to use large data sets to graphically model trends over a long range, the use of technology (perhaps excel) would be appropriate. In terms of substitution, using excel would replace the typical "pen and paper" route of graphing a handful of data points. Although the function is still the same, the program would allow students to represent data sets that are much larger and more representative of real-life application (i.e. water-hardness of drinking water tested monthly over the last ten years). This not only adds a new element to a previously completed task (augmentation), but also allows for significant redesign (modification) and redefinition of the task. The use of technology redefines the task in terms of content, but it also redefines the way students are able to approach the content. This leads me to my next point:

Engaging Disinterested Students 

I personally believe that atop the infinitely long list of "pros" regarding the use of technology in the classroom, sits the ability to engage disinterested students. The students of 2016 have been brought into the world with technology at their side. Having three nephews myself, I see that technology is such a powerful tool to gather their effort and attention. The conspicuous sound of a normally rowdy trio practically disappears when a  "Mindcraft" YouTube video gets played on the computer.

Visualization of disinterested students engaging with classroom material through the use of technology
- Image created on MS paint.

I think of all of the disinterested students in the classroom that continue to be pushed along the education system sitting at the back of the room, giving and getting little attention...Maybe they have difficulties socializing with others...maybe they are bored...maybe the pen-and-paper method of learning just doesn't click with them...Whatever the reason, I personally believe that the use of technology is without a doubt the most effective way to engage these students with the mathematics curriculum.

The integration of technology into the classroom could potentially challenge the way schools and classrooms operate at a fundamental level...but isn't that a good thing?

 “Many schools are organized as they are because they always have been, not because they must be.”
-Sir Ken Robinson

Thank you for reading,
Kevin Lavallee

Chapter 5: Classroom Community and Diagnostic Assessment

Positive Norms

This week in class, we discussed various positive norms to encourage in a math class. I just wanted to briefly discuss my perceived importance of the norms developed by Jo Boaler in this weeks post:

1) Everyone can learn math to the highest levels
2) Mistakes are valuable
3) Questions are really important
4) Math is about creativity and making sense
5) Math is about connections and communicating
6) Depth is much more important than speed
7) Math class is about learning not performing

Individually, each of these norms aim to develop a positive relationship with mathematics, however, as a whole, they aim to develop a sense of community in the mathematics classroom. I believe that this is critical for students to reach their full potential. Consider a classroom where students feel comfortable asking questions, making mistakes, thinking outside the box, offering help or opinion, or taking their time to understand a concept before moving on... Now consider a classroom where students do not feel comfortable to do any one or more of these positive norms. How can we as educators expect our students to succeed without effectively introducing and consistently encouraging these norms? Setting and promoting an inclusive mathematics community should be a top priority for all math educators.

Diagnostic Assessment

It is very important for educators to understand what prior skills/knowledge their students bring with them to the table.  This week, we took a look at some sample questions math teachers can use to assess not only prior knowledge, but also preferred methods of reasoning. The diagnostic question we analyzed was called "Making Juice". Essentially students were asked to decide/rationalize which of 4 cups of juice was the most concentrated. The 4 cups were as follows:

1) 2 cups of orange, 3 cups of water
2) 1 cups of orange, 4 cups of water
3) 4 cups of orange, 8 cups of water
4) 3 cups of orange, 5 cups of water

First, we were asked to solve the question on our own. I used fractions and percentages to solve that the first cup was the most concentrated with a value of ~66.67%. Next we were asked to examine some student solutions. The student work I examined used circle graphs in order to visualize the answer. His/her visual depiction was as follows:

Sample Student's Work - Image created on MS paint

He/she said that it was clear from the pictures and fractions that 2/3 was the largest fraction. Although the student is correct, the rationale is wrong. Looking at the percentages, 2/3 and 3/5 are very close, and may be very difficult to distinguish using a hand drawn manipulative. If we examine the students visual, he/she did not represent 3/5 in an accurate manner...in fact he/she showed that 4/8 and 3/5 are the same or roughly the same value. 

For me, the activity really opened my eyes to several things:

-the amount of thought that is required to develop good test questions

-how much more important the rationale is than the final answer (think journey vs destination)

-how much one question can tell you about a students prior knowledge and method of reasoning

Each of these "takeaways" are so important to remember when assessing students not only diagnostically, but also in formative and summative situations. Consider the student work above...the student was able to get the correct answer without understanding how to properly solve the question. If this question were multiple choice on a test, the student would have gotten 100% for a question they essentially modeled incorrectly. Although their journey led them to the destination, they are just as far away from when they began. Who knew that something as simple as asking a student to provide rationale for their answers could yield such powerful information?

Relating diagnostic assessment back to positive norms, it is important to keep in mind that your students are very good at "picking up" on what you value as a teacher. For instance, if your diagnostic assessment on the first day of class is a formal test, you are setting the tone for the community that doesn't necessarily reflect well on the positive norms...remember, consistently encouraging the positive norms is just as, if not more important than introducing them.

Chapter 4: Differentiating the Math Class

As a prospective mathematics educator, one of my biggest worries is ensuring that every student, despite their unique learning style, will be able to engage in the material I present. This week in class, we took a look at differentiated instruction (DI), which aims to achieve this goal:

"Differentiated instruction (DI) is adapting instruction and assessment in response to differing student interests, learning preferences, and readiness in order to promote growth in learning." (Student Success Implementation Branch – 2015, page 1).

Example of DI in the math class: Discovering and teaching linear relationships using a variety of representations (i.e. equations, manipulatives, graphs). See the image below:

Three ways of representing a linear relationship: Image created on MS Paint

In class, we read a ministry document called "Knowing and Responding to Learners in Mathematics". Essentially, the document provides insight on differentiated instruction
in Mathematics, through various classroom scenarios which "describe how teachers assess,
plan and adapt their instruction to determine and address their students’ interests,
learning needs and preferences" (Student Success Implementation Branch – 2015, page 1). In groups, we were asked to critically analyze one of the three classroom scenarios in order to identify any forms of differentiated instruction or formative assessment. As a class, we shared our results.

After reading my assigned document, I have to admit, I was feeling very overwhelmed. I though to myself "Wow, Mr. Young is an absolute superhero of a teacher". It seemed as though every single portion of his lesson was differentiated for every student. Thinking of all of the planning, pre-planning, adapting, and predicting that went into this one lesson felt like a very daunting task. When we combined our discoveries as a class, it was clear that all of the lessons in this document were "level 4" examples differentiated lessons, which only increased my sense of being overwhelmed.

Lucky for me, this feeling was eliminated during the debriefing portion of this task. Essentially, we discussed how important it is to use DI in the mathematics classroom...we want to provide all of our students with an unique/optimal avenue to display their knowledge/understanding, however, we also discussed that DI should reach the entire target audience over the course of a unit, not just a single lesson. Providing differentiated instruction is critical to the success of our students, but this does not mean that every single lesson has to be differentiated for every single student. This was my takeaway message for the week.

In short: The classroom scenarios found in the "Knowing and Responding to Learners in Mathematics" document provide amazing sample lessons which target effective differentiation strategies. Although this is an amazing resource, it may leave educators feeling a bit overwhelmed. It is important to note that effective differentiated instruction does not mean that every single lesson has to be differentiated for every single student, but rather it should reach out to each student's learning style over the course of a unit.

Thanks for reading!

Chapter 3: From Patterns to Algebra to Critical Thought

The traditional mathematics classroom, although never my absolute favourite, worked for me. Jumping through mathematical hoops became routine by the time I reach Brock University. For most math/physics/chemistry questions at the high-school level, I had a simple yet effective method for "retrieving" the correct answer:

1) Write down the information given. I always called this my "What do yah know?" section. 

2) Write down what the question is actually asking for in the end. This was my "What are yah tryin' to find" section. 

3) If it applied to the question, write down anything you needed to solve to go from given information to the solution. This was my "What do yah need to know to get there" section.

4) Use the information given to select the proper formulas required for the question. This was my very cleverly named "What formula are yah usin'?" section.

5) Solve all equations required to isolate the final answer. This is my "K done" section.

Jumping through Mathematical Hoops: Image created on Microsoft paint

More often than not, this was my fool-proof rinse, wash, repeat cycle for any math related question. Using this method taught me how to efficiently solve mathematics equations without ever really understanding the concepts behind them. Why would I want to learn more than I had to? I retrieve the correct answer and received my mark, and that was good enough for me. Of course there were exceptions to the rule throughout my high-school career, times where I really had to put the work in to understand an abstract concept in order to get the correct answer, but for the most part it was steps 1-5.

I thought to myself "Wow, I could probably just get through my entire Chemistry and Mathematics undergraduate program doing this". 

Cut to Thistle Hall room 245 (I think). It was late September, early October the first year of my undergrad. Everyone in the MATH 1P05 class was eagerly (or nervously) waiting for their name to be called to the front to collect their marked midterm 1 test. 

It was late September, early October the first year of my undergrad that I realized I could no longer play the game. University claims that they increase critical thought in their students, and I honestly believe in that claim. Whether it was math, chemistry, history, or education, I always found myself critically analyzing my every move. In university, I really put in the work to understand what I was doing, why I was doing it, and what effect  would it have on surrounding factors. It was a huge paradigm shift in the way I approached any problem, whether it be on paper or in my personal life. 

You're probably wondering where I'm going with all of this...

This week in class, we looked at the use of manipulatives and patterning to express linear relations. The concept blew my mind. It made me look at the concept of linear relations in a whole new way. I was trying to use patterning in my head to model a million different other concepts at once. My entire life, I literally trained myself to look at a problem from one and only one perspective. I thought I was being efficient, but aside from retrieving the correct answer, the only thing I became efficient in was mathematical ignorance. 

This activity made me truly understand the importance of:

a) using manipulatives in the classroom

b) teaching a concept through several perspectives

Using these two methods will allow all students to connect and engage with the curriculum material in a way which suits them best. As educators, we need to challenge the students who think they know the answer to look for it again without their set of "rules". I think this is the key to helping students really understand how/why something works at an early age. The world demands critical thought...this is how we provide it. 

Thanks for reading, 

Kevin Lavallee

Chapter 2: The Importance of Verbs (Week 2 Reflection)

What is a verb? Verbs express actions and states of being, and are arguably the most important part of a sentence. Without verbs, a sentence would just be a bunch of words! Lets try:

What a verb? Verbs actions and states of being, and arguably the most important part of a sentence. Without verbs, a sentence just a bunch of words!

See? Important.

This week, we discovered that the grade 9 mathematics curriculum has no shortage of verbs. During the first portion of our activity, we highlighted all of the verbs within the specific expectations of a mathematics unit. This took quite a while, even with breaking the unit into sections for each of us in the group. In order to visualize the occurrence rates of each verb, our group graphed the results (see the figure below). This activity "highlighted" just how many, and how often each verb was used.

Visualizing the "verb count" was overwhelming, and led to the questions of "why?", "what does this mean?", and "What implications does this have on me?".

Why? Well, it should be rather obvious that the writers of the curriculum place a significant importance on verbs. They are there to provide educators with guidance.

What does this mean? During the debriefing portion of the lesson, we discussed how verbs describe the route of engagement to the curriculum  for students. Again, they are there to guide teachers through the curriculum and how it should be accessed by the students.

What implications does this have on me? This notion really resonated with me. When reading the curriculum documents in the past, the verbs tend to be almost subliminal as the content jumps to the forefront. For example:

"Determine, through investigation, the relationship for calculating the surface area of a pyramid"

For the me in the past, this may have simply read "teach the students how to calculate the surface area of a pyramid", which is completely wrong. Think of all of the different ways that message could be interpreted. What if I had just given a traditional lecture on how to calculate the surface area of a pyramid, then passed out a work sheet with several questions asking them to solve the surface area of various pyramids. What would the students determine? Where is the investigation? What relationships would they be able to determine? Most, if not all of the actual intentions of the expectation are lost by skimming the document for "content".

Final Reflection. The verbs describe how the students should be taught, not just what they should be taught. This lesson really opened my eyes to the importance of analyzing the curriculum documents, not just to understand what I am supposed to teach, but how my students should be engaging the content.

Chapter 1: Introducing Kevin Lavallée

      Hello everyone...My name is Kevin Lavallée and I am a prospective educator in the Concurrent Education program (Chemistry and Mathematics, I/S) at Brock University. Although chemistry is my first teachable subject, this blog will mainly be written through the lens of a math mind. Before I describe why I chose mathematics as my second teachable, I wanted to present you all with a quote from Ken Robinson, a British author, and an expert regarding education in the arts. He said "We don't grow into creativity, we grow out of it. Or rather, we get educated out of it". Although he speaks to the education of arts, I believe that this statement can apply to the education of mathematics. Ever since I could remember, mathematics has been an outlet for me to express my creativity. Whether it meant trying to solve a question in a different way from my classroom neighbour, or developing a game out of my math homework, or optimizing the size of the tree fort I could build given the limited supplies I found in the dumpster at lumber store down the road, I always found a way to give what I had learned in the classroom a creative purpose. Although my "Math Story" revolves around creativity, I find that most other's do not. Referring back to what Ken Robinson said, I believe that a significant portion of mathematics students have had their creativity educated out of them. They believe that math is a right-or-wrong, yes-or-no, factual subject with no room for imaginative thought. Well, I believe that there is room for creativity in the mathematics classroom, and I believe that it is up to mathematics educators to make their students believe it too. Identifying and understanding this notion only progressed my passion for creative mathematics, and as a result the subject was a natural choice for my second teachable. 
      As a prospective mathematics educator, my biggest fear is incorporating the entire mathematics curriculum while attempting to develop a fun, creative, and relevant mathematics classroom. I hope that this course will help me gain practical knowledge and experience to ensure that creativity is not lost at the expense of the curriculum, or vice versa. This blog will serve as a tool to track my progress throughout this journey! Thank you for reading!
-Kevin Lavallée