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

Lesson's With A Good "Hook"

Does Your Lesson Have Context?

When creating a lesson one of the important things a teacher needs to take into consideration is the level of engagement the students will have. Obviously, we want the students as engaged as possible. Now I know understand that many teachers find it necessary to have classes that they lecture and students take notes to help lay some ground work for the upcoming lessons. However, it is important for a teacher to always be critical of their methodology for delivering lessons. Furthermore, teachers need to make sure they are taking the time to get to know their students and decide what delivery would work best with this particular group of students.

When looking at the educators I follow on twitter I seem to see a growing trend of teachers becoming more theatrical in the delivery of their lessons. Below is a great example to check out:

20 Books BY Teachers, FOR Teachers to Inspire Your Teaching via #DitchBook author, @jmattmiller https://t.co/VEVScqRxr1 #tlap pic.twitter.com/5GPCQ9g5ZH

— Dave Burgess (@burgessdave) October 18, 2016

Retrieved From: capt_hook3.jpg

I wouldn't go as a far as saying I'm telling teachers reading this to buy a new costume for each topic and turn your classes into a fully interactive play (although that's kind of an interesting idea to for integrated curriculum with the arts). I am however, strongly encouraging teachers to look at how engaging their lessons are. For me this is can be as easy as devising a creative "hook" to your lesson to capture your students attention (i.e., Captain Hook)



In the bigger picture, the point I am trying to make is to make sure that lessons provide some kind of context to them. This makes is so much easier to answer the time old question "why do we need to learn this. " Adding context to your questions whether it's bringing in the latest viral video or making math questions about real life problems this question quickly begins to be asked significantly less.

Do you have a unique way to add context to your lessons? Feel free to add your opinions to the comments below.

Technology's Place In The Math Classroom

With the majority of my education having taken place in the 21st century it is easy for me to see what an integral role technology has played in the classroom. Almost as a minimum teachers are using projectors and slide presentations a daily norm in the classroom. But to what degree should or can the education system incorporate technology into the classroom?

I believe it is easier to address the what can be used in the classroom so I will do that first. Honestly, the options are quickly becoming endless. Initially cost and internet connectivity were combined a monstrous inhibitor to technology in the classroom. However, as technology progresses so does the ease of it's integration.  Numerous students have their own PED and are able to bring these to the classroom. This is just a starting point though; teachers are able to use free websites such as Padlet, Secretive, Google Classroom,  PollEV, Kahoo and countless others.

It is clear that teachers have many websites and programs that are easily accessible to them. The next thing to think about is which types/forms/means of technology should be used in the classroom? I think the easiest way to address this issue is to look at from the perspective of our student's success. If the technology helps create an inclusive environment, helps the student(s) achieve learning objectives or aids in maintaining the engagement of the student(s) I believe it is easily justified to be used.

Another way we can look at this is presented through this article from EdTech; a website that is based around the use of technology in an education setting. I recommend at least taking a look at this article but the website as an entirety is fantastic.

In conclusion; I truly believe that incorporating technology into the classroom can help a number of students. However, one important thing to remember is that technology should always be used as a compliment to the teacher not instead of the teacher.


Feel free to let me know how you use technology in the classroom or any of your opinions on the use of technology in the classroom. You can do this by commenting below

EDIT:

I found an awesome infographic today on one of my favourite Tech Based educator's twitter feed. Alice Keeler suggests that we should be moving anything that isn't better on paper to a digital platform as a means to upgrade it. I wouldn't go as far as saying that you should be filling out this flow chart every time you make a lesson and moving all of your content online. However, I do think it should be in the back of teacher's minds to be critical about the means used to deliver a lesson.  Thinking about whether this could, as well as should, be done through a digital platform is always an important Especially as we move more and more into the digital age. Check out the infographic below:

How Is This Better Than Paper? https://t.co/x4Bl6HEDzP pic.twitter.com/am8aeuEWW7

— Alice Keeler (@alicekeeler) October 31, 2016

Cheers,

Mr. Studenny

Gaps In The Classroom

Students often find lessons difficult to understand or keep up with. I'm sure a lot of my readers can look back on their time in grade 9 or 10 math and remember feeling totally and completely lost. That feeling when the teacher is talking about something and you have literally no clue what they mean. That feeling, if your like myself, leaves you stressed, anxious and incredibly uncomfortable. It is almost as though you are missing that one piece of the puzzle to complete it.

At this point of your read I encourage you to go to the following google drawings page and attempt to get get all the blue shapes inside the black square. Attempt to do so without manipulating the shapes orientation or size or using the red square to complete. (EDIT: Please when you're are done return the shapes to their initial positions at the top, not necessarily in the order they are right now)

You will quickly find that without the red square it is impossible to fully complete this challenge.

You can think that each of your teachers are giving you one piece of the puzzle that will let you create a final product. However, imagine one of your teachers fails to successfully give you the piece. No matter how much you try you can't complete the puzzle.

One of the biggest struggles as a teacher is to determine where students gaps in their knowledge and understanding is and how to combat this. This struggle with putting the puzzle together gives a small insight into how students struggle when they have gaps in their education. With this in mind it is entirely reasonable for these students to not be able to meet the standards in classrooms.

So how do we determine which students have gaps and what those gaps are? 

This is defiantly the big question for teachers with regards to students who are underachieving. There are a number of techniques that teachers can use to help identify these gaps:

1. Diagnostic Assessments: before diving into new material taking the time to establish a baseline and make sure students understand all prior material that is necessary to move forward. 
2. Observing Conversations: a carefully trained ear can allow a teacher to pick up on subtleties in vocabulary and other communication skills that suggest gaps. 
3. Analyzing Answers: working through a students thought process can tell the teacher a lot about how they approach problems and what their gaps or strong suits may be. 
4. Ministry Support: the MOE provides resources to help address gaps students face. These can be found through the website Edugains. 

To conclude; it is imperative that teachers take the time to identify students who have gaps in their knowledge and furthermore, take the initiative to close these gaps. Through this process our teachers will be able to provide tools to certain underachieving students that will help them succeed.

---

Are you a teacher that has a strategy they use to close gaps? Feel free to share in the comments.
Are you a student who has faced a similar struggle? Feel free to share our story or insights in the comments.
Do you have any questions? I strongly encourage you to ask! 

Differentiation in The Math Room

In the classroom all of our students are unique and it is the teacher's obligation to treat and teach them as such. One of the more difficult aspects of teaching is making sure that your lessons are accessible to all students. The last thing you want to do is create barriers between your students and understanding.  It is easy to think of equal opportunity, accessible lessons and barrier free classes as taking the time to work with students who are achieving at a lower than average standard. This is can be problematic as you are no longer giving the attention to students who are gifted that they deserve. Take the following picture posted by @barrierfreemb on twitter as an example:

Equality vs. equity vs. the removal of systemic barriers #accessibility #DisabilityMatters pic.twitter.com/NBU3HwYfMU

— Barrier-Free MB (@barrierfreemb) May 12, 2016

The problem in this illustration is that the illustrated characters can't all see the ball game. In the first segment you see the characters each have an equally sized box; representing equality for all. Next we see that the boxes have been modified and individualized for each of the characters allowing them to all see; representing equity. In the final segment you see that the barrier that was put there in the first place being modified to allow all characters to see clearly without the need for boxes; representing the removal of systemic barriers.

When I look at this illustration I can so clearly see the connection to a key concept we've been recently introduced to that aims at creating an accessible lesson. The concept is to have your lessons built with a low floor and a high ceiling. This means that students who are achieving at a lower standard can access the same work that students who are achieving at a higher standard can.

One of the means of creating a low floor/high ceiling lesson is using techniques such as open ended questions as well as parallel tasks. Both of these aim to have students able to work on the same curricular expectations with the same task goals but at their own individual level. It is important to note this doesn't only benefit students at low and high standard achievement levels but also all the students in-between.

If you're new to creating open questions (like I am) it can be quite difficult to develop them yourselves. The important parts of open questions are that any student can access them.  An example could be asking students:

"How many apples are too many for a family"

instead of:

 "You are part of a family of 7. You each eat one apple per day 5 days a week. Apples last for one week before they go bad. How many apples should you buy for your family?"

Having this open ended question allows students to make it personal as well as control the complexity of it. One student may think of their family and easily say 20 apples are too many for my family. Another student may look at it more in depth by deciding how many people are in the family, how many apples they eat a day, if everyone eats an apple, etc. 

Marian Small, is very experienced at developing these type of questions. It is defiantly worth following her on twitter and her "good question of the week."

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Feel free to comment if you are currently working to use open questions in your classrooms.
Do you use them in math or other subjects?
What do you do to break down systemic barriers in your classroom to ensure you lessons are accessible to all?
What are you thoughts on equality vs equity?

Using Manipulatives for Pattern Recognition in Mathematics

Helping students understand the relationship between patterns and algebra can be an abstract progression. Using manipulative to help students visualize the patterns can be an incredibly useful tool. If teachers can start getting students to visualize the progressions of certain patterns and develop their skills in visualization the transition into algebra will be much smoother.

The following pattern is an example of this type of relationship:

Photo by author.

When I look at this pattern it is easy to develop an algebraic expression to determine the relationship. The expression n+4, where n=term number represents this.  Although it is easy to develop this expression it is important to take the time to look at this relationship and see why this expression makes sense. If we can have students focus on the fundamentals of developing this expression. If we can have students look at the diagrams and determine which parts are growing and which parts are consistent they can easily see relationship. Recreating these patterns with manipulative gives students the opportunity to visualize the progressions. The green triangles remain constant at 4. The purple square increases by one for each term; starting at one. 

The exciting part of this is that with repetition and practice students will be able to identify more complex problems and how they relate to linear relationships. 

Photo by author.

After practicing pattern recognition and developing algebraic expressions students can begin to relate these activities to graphing linear relationships. Take the pattern to the right for example. While recreating the patterns students will be able to develop the algebraic expression to be 3n+4. The difficult part in this example would be seeing that the centre square is also part of the fixed triangles. It would be through the practice that students would become accustom solving this. 

Photo by author.

From here, students can use manipulatives from the patterns on graph paper to visualize the growth.  Next they can track the progressions and relate the this growth back to the algebraic equation. This lays the frame work for discussions surrounding graphs as well. The relation between term 0 and the y-intercept is is easily seen. Extrapolation is also evident while using this method. 

Why Are WE Scared To Estimate?

My instructor brought up a great point the other evening about students struggling when asked to estimate answers to basic mathematic questions. (L. Surrturmm, personal communication, September 14, 2015) This really got me thinking and the more I thought about it the more I could relate to the issue at hand. Through this post I hope to shed some light on the reasoning and a possible strategy to correct this.

retrieved from: http://images.clipartpanda.com/estimate-clipart-math_estimate.gif 

I want you to think back to an experience I'm sure most of you all had in a math class. We've afll been given a math question asking us to first estimate the answer than compute it after to see how close we were. For example:

Question:

"If Jenny gets  $.47 for every Apple that she sells how many dollars while Jenny have if she sells 30 apples per day and works 90 days in the summer?"

Answer:
Estimation:___________
Computation:_________
Show your work:

_______________________________________________________________________

Now if you were like me I'd be more concerned about making sure my estimation was as close as possible to the correct answer because I was worries that the teacher would think I didn't have a good understanding of the content if it was far off. For that reason my approach to this question would be to fill out the show your work section first, then the answer. Finally I would put a number not far off of the correct answer as my estimate.

Looking back on this and trying to understand why I'd (I wanted to say children but that wouldn't be accurate because I'm not sure I'd be any less of a culprit to than I was back then) approached this question this way I came up with a couple reasons:

• I was caught up in the idea that I always had to have the correct answer.
• I was worried the teachers would be disappointed in me if the estimate was far off.
• I didn't understand the importance of developing strategies related to estimation.

Now that I've given this a bit more thought with an education, development lens I'd like to explain an option of how I'd have wished I'd approached this question:

First I'd highlight the important parts:

"If Jenny gets  $.47 for every apple that she sells how many dollars while Jenny have if she sells 30 apples per day and works 90 days in the summer?"

Next I would round the numbers to something more manageable:

-90 days in the summer would become 100 days.

-30 apples per day could remain 30 apples/day.

-$.47 for every apple would become $.50/apple.

From here I would say 30 apples/day for 100 days is 3000 apples. Then I'd think if Jenny makes 1/2 of a dollar per apple she would have half as many dollars as she sold apples. That would be 3000/2 which is an easy 1500. 

Making my estimation become; $1500 for Jenny over the summer.

________________________________________________________________________

If we look at this suggested process for estimation it resulted in an answer that was only a few hundred off and a 15.4% error - which if we consider the fact it took hardly any mental math to solve is pretty impressive. 

If we take the time to encourage students to think of ways to answer simple questions, like this apple problem, through basic reasoning and utilizing estimation we will be developing their adaptive learning skills.  In my opinion this is one of the best ways to set the foundation of have students thinking outside the box and develop creative solutions to complex problems. 

Our first steps to move students in this direction is slowing down our lessons and providing basic techniques creating manageable problems. We want to give the students the opportunity to make educated guesses. Once students start developing some skills and strategies for estimation then introduce more complex, challenging problems. 

________________________________________________________________________

Feel free to comment and let me know you opinions on estimation.

• Did you have any experiences similar to the one I outlined?
• Did you have an experience very different from this one?

Intermediate/Senior Physical Education and Mathematic Education

The posts I have been uploading thus far have been based around education in general.  As an intermediate/senior teacher candidate with teachables within Mathematics and Physical education I figured it would be beneficial to introduce myself in regards to these.

First of all; my name is Mr. Studenny and I am a teacher candidate at Brock University's Faculty of Education.  My first teachable is in Physical Education and my second teachable is in Mathematics. As a teacher candidate I plan to bring a combination of the skills I have learnt through these domains to the classroom.  An area that has really sparked my interest through my undergraduate degree has been in creating an engaging inclusive environment for my students. I find that the creation of a sense of community in the classroom can totally change and shape the learning environment in a very successful way.

I am particularly interested in how students react positively to a classroom that has a strong sense of community. I feel this helps engage and motivate students. Furthermore, it also sparks initiative and creativity in them. I view each an every one of these traits as a fundamental important characteristic of an engaging, successful classroom.

I can not wait to apply these principles to the physical education and mathematics curriculum.  I completely understand the struggles these will bring during the next few years of my professional life. I'm excited to face these challenges and see where they take me.

Cheers,

Mr. Studenny

Get To Know Your Students!

Retrieved from: https://s-media-cache-ak0.pinimg.com/736x/f5/57/5c/f5575c334502af9e19807b4b0e903cd1.jpg

As some of my readers know I am a prospective teacher currently in my fourth year at Brock University studying Physical Education. Through my time at Brock we are continually encouraged to design lesson plans with a "pre-test" at the start of this. The theory behind this is that by utilizing a pre-test a teacher will be able to quickly gather an understanding of the student's prior knowledge, experience and expertise on the subject. I believe these pre-tests are incredibly important and allow a teacher to make any last minute refinements, simplifications or extensions to the their lesson plans.

This all makes perfect sense; you don't want to be teaching a class how to add by twos when all the students can already do so at ease. The ideal lesson content for a student will be within their personal zone of proximal development.  Through this post I want to focus on the importance of working towards a deep understanding of where each of your students come from.

Why is it so important to get to know the individualities of your students? 

Each student in your classroom will come from a different family, in a different part of town, had different teachers, been on different teams and have so many other traits that makes them unique. To think that your lesson plan will work for each and every student is naive at best. As teachers it is our responsibility to get to know our students and adjust our teaching methods to best accommodate them. Having the ability to adjust the curriculum the match the needs of diverse students and the ever-changing issues impacting their world is a skill all successful teachers must have.

Drake. S., et al. (2014) suggest further depth to the individuality of a student. They suggest that students vary in learning styles, exhibit multiple intelligences and have preferences for their learning environments.  Each of these variables have had ample research done to determine which are best for students to learn. However, it is not about which is best for students but which works best for the students you have. It is all about actively deciding on a structure to your lesson that will work best for the student.

How can you get to know the individualities of your students?

There are many ways to get to know the the background information of your students. The first step is acknowledging that this is part of being a better teacher. This will help you connect to and understand your students much better. The following is a compilation of sources you many utilize to gain knowledge of your student's backgrounds as described by (Drake, S., et al. 2014):

• Review official school records, including current and previous report cards.
• Consultation with parents.
• Consultation with previous teacher(s)
• Consultation with support team.
• Classroom observation checklist.
• Educational assessments (pretests)
• Multiple Intelligences Survey or Learning Style Inventory
• Work samples, assignments, and projects.
• Portfolios.
• Teacher-student conference
• Peer and self-assessments.
• Interest surgery. 

Each of these sources of information go hand in hand with being a good teacher. It may seem like an incredibly daunting task to gather and comprehend all this background information on students but it will definitely pay off. This is the different between knowing that that kid in your class isn't participating because he hasn't had breakfast at home and all you have to do is give him an apple compared to dismissing him as lazy. 

Where do you go from here?

Gathering this background information is not the end of the story for teachers. The profiling and understanding of their students is an ongoing, developing process. From here it is important that teachers continue to seek out those personal pieces of information to make sure they understand what is going on in the child's life. Soon, as a teacher, you will be able to know your students well enough to judge their moods and determine when something is just off about them. This then translates directly into each and every lesson you have planned. Some days you may need to do something totally different than planned to accommodate the mood or stresses your students are dealing with. 

My point is that being able to get to know your students and be able to judge their moods and understand them at a personal level will make you that much better of a teacher. It will make you a teacher who is able to connect and influence your students lives in the ways they need you to. 

Until next time,

Mr. Studenny 

Drake, S. M., Kolohon, W., & Reid, J. L. (2014). Interweaving curriculum and classroom assessment : engaging the 21st-century learner. Don Mills, Ontario : Oxford University Press, [2014]

What type of student you were/are? :). JPG. Retrieved from: https://s-media-cache-ak0.pinimg.com/736x/f5/57/5c/f5575c334502af9e19807b4b0e903cd1.jpg

Approaches To Instruction | Developing Your Teaching Identity

In a few years time I hope my aspirations of becoming either a physical education or mathematics teacher materialize and I have my very own classroom. Through my time at Brock University thus far, I have been introduced to the importance of developing my "teaching identity." This is a broad term that encompasses the importance of understanding your values, stances and beliefs of practices in the classroom. Furthermore, it is address the understanding that there is not one set method of instruction to manage a classroom effectively.

I believe that all prospective teachers need to address this idea of developing their teaching identity.  It is worth carefully thinking about various aspects of teaching and what your views are on them.  It will be all prospective teachers responsibility to develop a stand point on issues that arise in the classroom and educational world.  These issues could range from how to properly discipline students to the best way to arrange seating in a classroom.  Today I would like to address the various approaches to instruction. I view the choice of methods of instruction a primary concern for all teachers.

Studenny, Mike. "Approaches to Instruction." 2015. JPG.

There are many approaches to choosing a method of instruction. I have developed the following graphic organizer to show a variety of different approaches to instruction as described by Drake, Reid and Kolohan (2014).  As you can see the organization of instruction methods is circular. I have attempted to convey the message (one I strongly believe in) that these are not ranked through a hierarchy but rather viewed as equals. By this I mean that a teacher should understand that each method has it's place and uses. This video (Long-Crowell, E. 2015) demonstrates a good contrast between the values of direct instruction and the guided discovery model.

Now I would like to get back to the idea I presented on developing your own personal teaching identity. I believe every teacher has a bias to which instructional approach they can best utilize in the classroom. This bias may have developed from you being taught this way or possibly you having seen good results from it thus far. Either way as prospective teachers we need to cast these biases aside and realize that students have their own individualized learning styles (Advanogy, 2015) and therefore find will  find various approaches of instruction may work better than others.

So, if a teacher is able to ignore any biases to instructional approaches and really understand that each student will have an individual learning preference how do we cater to all of our students? I propose that we view these approaches to instruction as a spectrum and focus on the importance of building a blend of many approaches into each and every lesson we teach.  It is through this progressive approach to building your classroom dynamics that a teacher will begin to address more individualized needs of our students.

References

Advanogy. (2015). Overview of learning styles. Retrieved from: http://www.learning-styles-online.com/overview/

Drake, S. M., Reid, J. L., & Kolohan, W.  (2014). Interviewing Curriculum and Classroom Assessment: Engaging the 21st Century Learner. Don Mills, ON: Oxford University Press.

Long-Crowell, E. (2013-2015). Direct Instruction and Discovery Instruction: Definition and Differences. Retrieved from: http://study.com/academy/lesson/direct-instruction-discovery-instruction-definition-differences.html

Studenny, Mike. "Approaches to Instruction." 2015. JPG. 

Constructivism And Normal (Bell) Curves.

Bell curves. I remember first hearing about them in high school. My rough understanding was that a teacher would be able to look at any group of grades and, hypothetically, produce a curve where the majority of students will receive similar grades. However, this means that the other-smaller group of student's- grades would be equally split between being lower than average and above average. During my time in high school the practical aspect of this bell curve was that if the majority of students didn't do well on the assignment the teacher could just move everyones grade up in proportion to this normal curve i.e.. bell curve the grades. As an adolescent this idea made sense to me and I believed it was beneficial to the students as it made up for any discrepancies between the teachers methods and the grades you deserved. 

Enter constructivism. I was first introduced to the ideas of constructivism during a first year university education course. I remember the explanation that resinated most with me was that constructivism is an ongoing partnership where the teacher helps the student build their own understanding.  At that point constructivism-and later on the Socratic method (Scarince, 2015) as a teaching tool-made so much sense to me.  As I have such a big interested in athletics I have a strong grasp on that doing something is a great way to learn it.  Another great explanation of constructivism by Drake, S., Ried, J., & Kiloton, W. (2014) "if learners are active, self-motivated builders of their own understandings, all students can learn and succeed...". 

Studenny, Mike. "Bell Curves in the Classroom." 2015. JPG.

With that quote in mind I'd like to re-evaluate those bell curves, their implied principles and their place in "traditional" teaching. Having a classroom that is based around the expectation that grades will be distributed across a normal curve means that the majority (68.2%)of your students will get grades that are considered average, standard or normal. Furthermore, this model predicts that 3.2% of students will excel and achieve high standards.  So far this model seems to be looking pretty positive for the classroom. However with looking a little deeper you can see that although 3.2% are achieving a high standard an equal amount of students are at the lowest level as well. Personally I believe any teacher that has student's falling in this section should focus on their lessons and teaching to accommodate these students. But what about that 68.2%? Doesn't having the majority of the class achieving average grades classify as successful teaching? No. Every single teacher should have an approach that aims for each student to achieve the highest standards possible. A teacher should not be satisfied with mediocracy but instead strive for exceptionality. 

I believe gravitating away from a principle that is based on bell curves and towards a more constructivist based learning classroom students will begin to transfer forwards on the curve and join the 3.2% who are achieving a high standard. It will be this idea of students learning by building their understandings and knowledge that pushes them forward and down the curve to this area. 

References

Drake, S. M., Reid, J. L., & Kolohon, W. (2014). Interweaving Curriculum and Classroom Assessment: Engaging the 21^st     Century Learner. Don Mills, ON: Oxford University Press.

Studenny, Mike. "Bell Curves in the Classroom." 2015. JPG.

Hello and Welcome!

First of all I would love to extend a warm welcome to any new readers visiting my blog. This blog will be my very first attempt at publishing myself online and I'm honoured that you're here to be a part of it! As of this post, I am a fourth year student at Brock University in the Concurrent Physical Education program with a second teachable in Mathematics and am very excited to be start sharing my thoughts, opinions and experiences with all of you. My goal, and plan, is to develop this blog by addressing issues related to education and analyzing my own experiences in the world of education. I hope you enjoy my posts and share your opinions with me through comments.