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!