Adding Some Mystery To The Math

This week we were given an interesting take on an atypical math lesson. My peer presented a lesson themed around an Escape Room. For anyone that's unaware, these are becoming increasingly popular and are physical rooms that you and a team need to solve a puzzle to break out of the room.

Retrieved From:  http://www.meenagames.com/wp-content/uploads/thumbs/custom/M/Math-puzzle-room-escape-game.jpg

The idea of using this theme for a lesson is both interesting and attainable for me. Finding ways to captivate students can be an ongoing challenge for teachers. This is definitely an adaptation to the typical classroom environment that is sure to spike the interest of students. This would easily be considered a lesson that reaches a variety of different instructional needs for students. Furthermore, it creates an environment that is prone to be at the optimal challenges level for students (i.e., the zone of proximal development.)

This lesson drew on such a wide range of understandings, math concepts and also very importantly math processes. This creates an environment that encourages students to work as a team. From here, the students are able to each contribute their strengths to work together to solve the puzzle. This allows students of varying abilities to share their expertise and each find success in their own way.  Being able to structure a class in this way, this successfully is the representation of an excellent educator. I will strive to bring this expertise to my classroom one day.

Sinusoidal Spaghetti

When I was first learning about sinusoidal functions we were taught using a very direct teaching method. My math teacher showed us the equation, the graph and a number of key features of the function. The teacher then compared and contrasted these portrayals of the function, all the while the class (and my self) copied down the notes. The way I retained this knowledge was through reviewing my notes and slowly memorizing the function and it's properties. 

Retrieved from: https://mathbitsnotebook.com/Algebra2/TrigGraphs/phasepic5.gif 

During our course this week I was introduced to a student centred learning model for the sinusoidal function. My peer introduced this activity as sine wave spaghetti. We were given a bit of review of content the students would need to know before attempting this activity. Then we were partnered up and worked as a team to co-create the sine wave. We used a cartesian plane with a circle thats radius was 1. From here we found the sinx=opposite/hypotenuse values for a many points along the circle. We cut pieces of spaghetti to this length. Then we placed them at corresponding locations on a graph. Finally we drew out the lines to show the representations. 

This activity would have really helped me grasp the relationship of the sinusoidal function if I had been given the opportunity to do it as I learnt functions. As an educator it is important to explore various ways of representing content to students. This is a direct example of differentiating instruction and how it can benefit a number of students. 

A Hands On Approach To Maximizing Area

This past week in my teacher's education mathematics course we looked at teaching the applied stream of mathematics at the high school level. Particularly we looked at the grade 9 and 10 level.

A struggle a number of educators face is the fact that the age old saying of "practice makes perfect" really does ring true for a lot of students. In mathematics specifically it is essential for students to be given the opportunity to practice and apply their knowledge and understandings. The difficult part here is that if the students are just given practice problems over and over they will generally respond with disinterest. The key to combat this is to almost hid the math in a more engaging activity.

In our class one of my college presented an activity that did exactly this. The activity was game-based learning where students were able to demonstrate their understanding of perimeter and area. The game was involved two students playing against each other on a large grid paper. The goal of the game was to shade in more space than your opponent. Before shading an area in you had to role a set of dice.The resulting numbers on the dice dictated the dimensions of the square the play would shade. Accompanying the game board was a chart that each player recorded their moves on. The progressions of this chart helped organize the students game play while visualizing the math they were using. Once the dice were rolled both numbers were recorded; one as length and one as width, then the area was calculated and lastly the perimeter was calculated. This resulted in a chart of a variety of combinations of rectangles with lengths and wides ranging from 1-6.

The valuable part of this activity, as I alluded to, the students are engaging in an almost hidden form of math.

Cup Stacking & Linear Equations

This week in my teacher education mathematics class we visited a problem of stacking cups. The problem was created in a few levels. First our instructor showed us one cup and posed the question of how many cups tall she was. We all provided our estimations. These were recorded to be revisited later.

Next we were asked what information we would need to find out an exact answer. We decided that we would need the height of the cup and the height of our instructor. From here it wasn't very difficult to solve for the number of cups. We chose to stack the cups in alternation of inversion. All we needed to do was divid the height of the instructor by the height of one cup. This would give us the height of our instructor with cups as the unit of measure. We compared this to the estimations to see which groups were close. This provides opportunity to discuss errors.

Our instructor then posed the next level of the problem; we had to stack the cups all the same way (how they came in their packaging). Now she asked what new information we might need to solve the question. We decided that we needed to know the height of the "lip" of the cup. From here the solve was a little more difficult than before but still easily comprehended. You would have the height of one cup (the top cup) plus x number of cups to reach the height of the instructor.

At this point the math gets a little interesting and the connection to linear equations becomes evident. With a group of students you can have them develop the equation that represents this linear relationship from only the information given so far. The y-intercept would be the height of one cup and the slope of the line would be the height of the "lip" of the cup.

Another approach to this problem could be as follows. Provide the students with two district heights of stacked cup (i.e., 5 cups is 20cm high and 15 cups is 32cm high.) From here the students could use the two points as coordinate points and develop a linear equation to represent it from here. This would involve them deriving the slope of the relationship with ∆Y/∆X.

Overall, this is engaging activity and provides easy and practical connection to linear equations. Furthermore, it is easily modified to allow for differentiated instruction which as educators know is fundamentally important to a good less.

Developing a Conceptual Understanding of Dividing Fractions

?Why do We Invert and Multiply?

If you ask someone how to divide two fractions they will generally respond with the timeless rule of "invert and multiple." When I think back to my math education this was exactly how they taught me to divide fractions. It was simply instructed as one of those things you need to memorize and will always work. To be honest, I have not given it much thought since then because I easily remembered this rule and that has been good enough. 

I have given a lot of thought this past week on the conceptual ideas of dividing factions, trying to wrap my mind around it. I tried not to look at literature or other explanations on building a conceptual understanding as a means to express this understanding as my own work. With that being said, I am sure there are many more complete explanations out there, but I do hope this can serve as a description of my basic understanding. 

I built a better understanding of this by going through a process of different division questions with increasing complexity and looking at them conceptually. The process went from dividing whole numbers by whole numbers to whole numbers by fractions to fractions by whole numbers to simple fractions by simple fractions. I am still struggling working with the conceptual representation of the division of complex fractions, but hope the ideas of a simple fraction can be extrapolated.

Dividing whole numbers by whole numbers.

Lets look at 4÷1=4.

For this equation we have 4 whole parts divided by 1.
OR
We are looking at how many times 4 can be split into groups of one.
OR
We are looking at how many times 1 can fit into 4.

With the purpose of conceptualizing division I find it makes the most sense to look at this through third example. It is easy to visualize this idea.

Dividing whole numbers by simple fractions. 

Lets look at 4÷2/3=6.

For this equation we have 4 whole parts divided by 2/3.
OR
We are looking at how many times 4 can be split into groups of 2/3s.
OR
We are looking at how many times 2/3 can fit into 4.

Figure 1 (Original Content by Author)

Still looking at the third example we are trying to visualize 2/3 fitting into 4 wholes. This is easily expressed using circles. Figure 1 shows that you need 6 2/3's to fill the 4 wholes. For this reason the answer is 6.





Dividing simple fractions by whole numbers.

Lets look at 2/3÷4=1/6

For this equation we have 2/3 divided by 4.
OR
We are looking at how much of 2/3 can be divided into 4 groups.
OR
We are looking at how many times 4 can fit into 2/3.

Figure 2 (Original Content by Author)

Still looking at the third example, we are trying to visualize how much of a 4 wholes will fit into a 2/3 section. Looking at it this way, we visualize the 4 wholes as one entity and the 2/3s as a second entity. We are trying to see much of the first entity fits into the second. I find this easiest to visualize using rectangles. Figure 2 shows that you need 1/6 of the 4 wholes to fill the 2/3s.

Dividing simple fractions by simple fractions.

Lets look at 3/4÷1/2=3/2

For this equation we have 3/4 divided by 1/2.
OR
We are looking at how much of 3/4 can be divided into groups sized 1/2.
OR
We are looking at how many times 1/2 can fit into 3/4.

Figure 3 (Original Content by Author)

Again, using the third example and the same ideas we have developed in the previous section, we can tackle this problem. We look at 1/2 as one entity and the 3/4 as a second entity. We are trying too see how much of the first entity fits into the second. Again, we can use rectangles to help visualize this in Figure 3; this shows you need 3/2 of 1/2 to fill 3/4 OR 1 1/2 of 1/2 to fill 3/4.




Concluding Thoughts

I hope to continue thinking through the conceptual models of dividing fractions and revisit this post to include an evaluation of dividing complex fractions by complex fractions that I am happy with. At the current time I have not been able to articulate it in a way I feel adds to this blog.

All in all, I hope the first few examples have provided a deeper understanding in to the why and how we divide fractions.

My First Teaching Practicum

I had the opportunity to complete my first teaching practicum in my primary teachable; Physical Education.  I was in charge of teaching a grade 11 open physical education course of all male students. With this being my first time teaching a course there were number of things that I didn't expect and even more lessons for me to learn.

Going into my practicum I thought that the time it took to lesson plan a lesson each and every day would be the most difficult part. I quickly realized how mistaken that was-yes the lesson planning was time consuming but soon I became accustom to it. In fact, the biggest struggle I found was developing and establishing an effective routine with the class. The student's themselves had a difficult time with maintaining a regular attendance of the class. This made it a struggle to implement any routine at all. I decide to re-think my ideal routines and begin to work through a routine that was tailored to the class.

My routine was based around always having a structured plan for the class but leaving opportunity to take the students feedback into the daily activities. We would start the day with 5 minutes of personal ball time; where the students could use any equipment from the days lesson to warm up and "play" with their peers. The students always responded well to this. From here I would make a judgment call to see if they appeared to all be sufficiently warmed up or if we should do a group warm up. After this we would have a class "huddle" where we looked at the daily learning goals and agenda. The students would then know what to expect for the remainder of the class. Going through the activities of the day the students would be allowed to suggest any modifications to the rules during the instruction periods. This allowed them to have a voice in the activities we engaged in. At the end of the lesson we would debrief and always discuss strategies and tactics used throughout the day; what they found successful what they had difficulties with. We would end this debrief with a review of whether or not they felt they achieved the learning goals from the day. I would use this debrief information to help guide my planning for the next lesson.

Having this routine in place created a structure to the class that I believe the students really appreciated. I could easily see them become more engaged in the lessons and in turn were reaching the learning goals more regularly.

It is for these reasons that I honestly believe one of the most important lessons I learnt during my practicum was how significant a well developed, personalized routine is for the success of a class.

Reflecting on Online Session 1 and 2 - EDUC 8F83

This blog post is being used as a reflection of two online sessions for my EDBE 3F83 course.

Session 1

In the first online session we dove into the problems with mathematical discussions in the classroom, or more accurately the lack there of. The article Orchestrating Productive mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell provides a great set of practises to apply to the classroom. These are:

1. Anticipating likely student responses to cognitively demanding mathematical tasks
2. Monitoring students' responses to the tasks during the explore phase
3. Selecting particular students to present their mathematical responses during the discuss-and-summarize phase
4. Purposefully sequencing student responses that will be displayed
5. Helping the class make mathematical connections between different students' responses and between students' responses and the key ideas. 

Using these strategies will help develop a classroom community that encourages and thrives from mathematical discussion. The one practice that really resonated with me is the first one. As a teacher, being able to anticipate students' responses - even the more abstract responses - will prepare you for a more in depth discussion. Furthermore, this will push your classroom into thinking about the problem  through a variety of lenses and ultimately gaining a deeper, richer understanding of the problem. 

Another aspect of this first online session that I found very important was the idea of developing mathematical curiosity. If we, as educators, can foster a curiosity in the minds of our students they will inevitably be much more driven to resolve problems. Finding creative, effective ways to do this is fundamentally important for the ever-changing classrooms we face. We are well aware of how individually different our students are, for that reason finding a wide variety of ways to spike the interest of students is ever so important. 

Session 2

In this session we looked into formative assessment and more specifically into providing feedback as learning.  One issue that continually arises in the classroom is the student who has not yet reached the correct solution to the problem. Often they will have taken a few of the correct steps but then somewhere, along the way, misstepped and arrived at an incorrect or incomplete answer. When assessing work it is important for teachers to address this issue in an effective way.  Looking back to the first online section and the five practices; practice one will help greatly with this. As a teacher we must attempt to follow the students path of work and see where there was a discourse. From here we need to develop feedback to help the student realize the problem and ultimately correct it. This is a lot easier said than done. 

As an educator it is important to keep a few things in mind when providing feedback. It should be positive, help the student diagnose their problem and provide direction. If formative feedback can follow these important features students who are on the cusp of a correct answer will surely self-correct and achieve their potential. 

In conclusion, these two online sessions have helped me gain a deeper understanding how to foster mathematical discussions in a classroom as well as provide beneficial formative feedback. Furthermore, it has shown me the incredible importance of being effective at both of these skills. 

Cheers,

Mike Studenny