Course Overview and Wrap-Up

It's crazy to think that we are already done classes.  I still remember going to my first class in August and I was thinking that March seemed so far away at the time.  In this class, I definitely learned a lot about effective teaching of mathematics, unit planning, lesson planning, spiraling the curriculum, backwards design and many other components that I did not list.  Overall, this class was very insightful and it provided me with comfort that I am ready to teach math in a high school classroom

Moving towards my next block, I feel a lot more confident in my ability then I did going into my first block.  Even when I woke up today for my observation day I wasn't thinking "how am I going to do this" like I was on the first observation day.  I feel that goes for everything that you do for the first time.  Whether it be riding a bike, going skiing, first university exam you are going to get nervous as you have never experienced it before.  I am teaching math in this block and I greatly look forward to it as I feel it will be a change from my other teachable geography.  I know I will definitely use lesson activities and skills that I learned from this class to help with my everyday planning.

If I could take one thing away from this course it would be that teaching is not an easy task (especially math) as there is so many different abilities within one classroom.  BUT, that being said it is an incredibly rewarding profession because once that student has that "Aha" moment and is able to understand the concept that you are trying to teach them then it makes you feel good.  It makes all the nights that you spent planning and changing your ideas all worth it when your students have a great time in the classroom and are still learning the concepts at the same time.  Math doesn't have to be chalkboard and textbook work all the time.  Why not use creative apps and games such as Desmos to enhance student's learning because let's be honest textbook work can get really boring day in and day out.  I know that when I go to my math teaching block I will try my absolute best to make my lessons as creative and technologically interactive as I can. 

Escape Room Activities in Education

This week in class we had a lesson activity created by one of our colleagues based upon grade 12 Advanced Functions.  He focused the lesson primarily on trig identities but the actual way he taught the lesson was one of the most innovative and interactive ways that I have ever witnessed.  The instructor set-up a complete "escape room" activity where we had to solve problems, figure out clues figure out famous mathematicians all to obtain a final key which opened a locked box.  While doing this escape room, I was having so much fun that I forgot I was even doing trigonometry at the same time.  There were so many twists and turns throughout the activity which allowed all the participants to really use their brains and think critically.  Overall, it was a great activity.  The video featured here has great ideas and tips to include in your escape room for your class.

https://www.youtube.com/watch?v=yqzczyx5dss

In terms of applying this activity to a unit, it can be seen that it could be used as a culminating activity or used as a study period before a test.  I feel that this should only be used near the end of a unit because you have to apply all the information you would've learned throughout the unit to solve the final riddle.  This is what makes it very useful for a culminating activity as you as the teacher can go around and observe how the students are interact with the problems and working together to achieve the answer.  Moreover, the idea of a "escape room" can be set-up just for fun before a test as it does review major concepts but takes the stress out of having a quiet study period in the classroom.  This may in fact not only help students learn what areas they need to still study but allows them to relax and have fun in the math classroom.  Debriefing the activity is a great way for the teams to reflect on the process, how well they worked together what they would've done differently, what areas did they struggle with, what areas did the do well with?

Finally, what I really think is great about this idea is that it can be applied to almost every subject in a highschool curriculum.  For example, my second teachable is geography and I know I could create a great escape room activity on something like Canada's resources and industries.  This could also be applied to a unit in English if students were studying Shakespeare for example as it can be a creative way for students to brush up on their knowledge.  The website at the bottom has great resources for you to check out and apply the escape room to your activity.

https://www.theescapeclassroom.com/

Grade 9 strategies to use in the classroom

Two weeks ago, we had a presentation for our "teach a lesson" assignment by one of our colleagues in class.  The topic was slope of a line and they created a very interactive game that was designed to engage students while they were learning.  Instead of just practicing boring slope questions from a textbook or assignment, the teacher candidate created a game called battle slopes.  This game was very similar to battle ships but combined the idea of slopes of a line and equations of a line  How this game worked was that each player picked a partner and was given a grid with an x-axis and a y-axis (each grid was hidden from the opposing player).  They then had to place dots which represent different sized boats (there was a 2 dot boat, 3 dot boat, 4 dot 5 dot boat) on the grid wherever they choose.  Next, much like the game of battleships, instead of calling out a co-ordinate like A5, they would call out a random slope of a line.  For example, my first move I called out 2x+3.  My opponent had to then draw that line on their gamecard, if it hit any of the points they placed, then they would declare "hit".  If it didn't hit any points, then my opponent would say "miss".  This would continue until one of us hit all of the battleships. 

    I thought that this was a very interactive game as I had lots of fun playing it but I was still learning at the same time.  Although I know how to plot the slope of the line, it was a good refresher for me and I could really see myself using this in class.  Another thing that the colleague did was have a mathematical discussion about the game after everyone finished.  We essentially all compared our different strategies to one another to see what the ideal method was to "hitting all the ships".  Some people (like myself) just more or less randomly guessed until they hit a point and then moved the y-intercept to hit another point.  Other people started with y=x, then y=x+1, then y=x+2 until they had the entire board covered.  We also discussed whether it was better to call out a steep line or a flat line when playing this game. 

Overall, going into my next block which will be a math teachable I really learned from this colleagues presentation.  I truly feel that they did a great job and this is definitely an interactive game that I could use in my placement near the end of the slope unit.  I feel that this would be great practice for students if they were studying for a quiz or a test or even if they needed clarification on the slope of the line.  Moreover, the mathematical discussion at the end was a great way to get everybody involved and see all the different strategies that one could approach this game.

 

Cup Stacking

This week in class we looked at a cup stacking mathematical question.  We were supposed to determine how many cups tall the teacher is.  This was an extremely open ended question and we first had to figure out what information we would need to start the question.  After careful consideration we found out that we needed to have the teacher's height, the height of one cup and the method of stacking.  We found out that the teacher was about 1.78m, the height of one cup was about 11cm and we had to interlock the cups.  From here there could be a multitude of different ways to go about solving this problem.  If this was in a classroom setting, I would leave this problem completely open to students and tell them to try it using any method.  My first initial thought was to physically stack the cups beside the teacher and try to get a rough estimate but we later found out that there was not enough cups to do this.  Eventually, we figured out that if we determined the height of each lid then we could just continue stacking until we reach 1.78m.  But, this was one only method.  Some students used linear relationships, graphing etc. which all can lead to the final result.
I really believe that these open-ended questions are great because they enhance a student's critical thinking skills.  This can also lead to "mathematical discussions" in the classroom in what we studied earlier in the year.  Students can go up in groups and show how they got their final answer and then can compare and contrast with other students creating a mathematical discussion.  These ideas are not only fun but they provide a rich experience for all students.

EQAO grading

     This week in class, we took a look at how EQAO for grade 9 math, is assessed.  An answer can be given either a 10, 20, 30 and 40 dependent on the mathematical quality and content that the answer has.  A level 10 essentially means that an application of knowledge and skills is limited and is caused by a misunderstanding of concepts or an incorrect selection of procedures (eqao.com).  On the other hand, a level 40 means the application of skills is at a high degree caused through a high understanding of concepts and an accurate application of the produces (eqao.com).  Although it seems relatively simple to give students appropriate grades for their questions answered we found out in class there can be a lot of difference of opinions between teachers.  There was one example where a student had slightly written the wrong formula for the area of a circle as A=pi^2*r but had all the correct steps and we had to decide on a grade for that question.  Some people in the class believed it was a level 30 while others believed it was a level 20.  This is where the difference of opinions came in and there was a lot of deep discussions as to what score we felt that student achieved.  Another example was a student who had the correct answer but didn't provide a concluding statement.  Some people in the class felt they deserved a 40 as they did achieve the right answer but others felt the student deserved a 30 as there was no concluding statement. 
     This small exercise that we did in class definitely had large implications on my learning.  First, I found out the difference of opinion that teacher's have when it actually comes to marking.  In most situations, there is only one teacher who does the marking for one specific class so it is usually up to their jurisdiction.  For the EQAO, there are several teachers who mark them and have to collaborative and decide on a final score for each student's questions.  We are continuing this exercise in the next class and we will compare our selection of grades with other members in the class.  Overall, this was a great experience as I understood the difference of opinions that teacher's have when it comes to assessing student work.

Consolidation: Online Session 1 and 2

Mathematical Discussions and their Importance:

Online session 1, dealt with the importance of Mathematical discussions in the classroom and ways that we can go about orchestrating them to create collaboration between students and the teacher.  I believe that one-dimensional questions with only one answer or method to solving them does not promote mathematical discussions in the classroom.  It is important that the math questions designed are well thought-out and are held at a higher-level of thinking.  This will prevent students from simply memorizing a formula or memorizing a method to answer the problem.  Making the questions open-ended essentially allows the problem to be solved in various ways.  For example, in a classroom, you could have several students employ different methods of how they achieved the final answer.  This collaboration and multiple methods of solving the problem will promote mathematical discussions between the students.  These types of questions make the students use their critical thinking skills which overall has them make connections and truly think about the problem.  We as a class did an example of calculating the number of patio tiles in the pattern given.  We were told to answer the question in three different ways and it really made me look at the question and think about different approaches to solve it.  Sure, the first and obvious way of solving it was easy but the other two ways did take some time and careful critical thinking.  Once we completed the question we opened up the discussion to fellow peers so we could compare and contrast our results.  This was a very relevant lesson and I can use the idea of open-ended questions in the classroom to promote mathematical discussions which results in greater cognitive development for the students.


Importance of Formative Assessments and Feedback

   Formative assessment is important in the mathematical classroom but this does not always mean it has to be written tests or quizzes for grades.  Formative assessments is still a form of feedback and can be portrayed in manner forms such as; exit cards, math interviews, observations etc.  They all serve the same purpose as they help the teacher gauge where their students are at with the lessons and content.  If a teacher notices (through formative assessments) that their class has been struggling with a concept or lesson they are then able to go back and review the material with their students. Essentially, formative assessments are important because it gives students a better understanding and helps clarify any possible questions that they may have had about the previous lesson or content.

   Feedback is important for students in a math classroom as it helps them improve their skills and their mental thinking about approaching a problem.  The way the teacher demonstrates or provides the feedback is essential for student's learning as too much feedback may be overwhelming but too little may leave the student feeling confused.  Individualizing feedback and finding an appropriate balance is a great way to provide solid feedback to all students.   Moreover, Feedback is extremely important for students who may be struggling in the class.  For example, if a student is struggling, you wouldn't want to just write a big red x on their test question as this may continue to discourage the student from the class. Instead, you may want to write one positive comment such as: "you were on the right track but...." and then provide some feedback on the question.  Completing the feedback in a timely manner is also important as it gives the students enough time to digest the information and truly learn from their own work.

Importance of Context in Lessons

Context is extremely important in everyday life as it provides us with the underlying reason for the idea.  Without context, we would often be confused during a conversation as there would be plenty of missing information and questions raised. For example, if I were having a conversation about the weather with a person and left out the location (or the context), they would be very confused as to what city I was referring to.  We learned this week that providing context in mathematics is extremely important to "hook" the students and answer the age old question of "why do we need to learn this?"
To begin, we looked at several open-ended problems in class and we attempted to tackle them in groups of four.  One of these examples we looked at was determining the number of possible toppings on a Harvey's burger if there was a total of 2048 combinations.  If this was a data management class, this would be a great introduction into combinations and permutations as it really catches the student's attention and gets them thinking.  Our group attempted to figure out the total combinations if there were one, two and three toppings and then tried to find a recursive formula using a table of values.  This was a good idea, but it was pretty time consuming as we had to make sure were weren't leaving out any possible combinations.  After this activity, we discussed the results as a group and found out that everybody tackled the problem differently but still achieved a similar answer of 11 toppings.  The context of this problem makes it a lot more applicable to everyday life compared to a boring old "black and white" textbook question.  I feel that providing context is important to keep the class engaged as well as provide an answer to "why do we need to learn this?"  In the Harvey's example, we actually touched upon Combinations and Pascal's Triangle which are both important topics in data management and statistics.  Essentially, even though we were trying to figure out the answer using a variety of creative ideas and methods we still brushed upon important mathematical concepts that are essential for the statistics unit. 

In relation to our class as future educators it can be seen that context can increase overall engagement in the subject and enhance critical thinking skills especially with open-ended problems.  The video attached to this blog shows a teacher describing her experience with cultural context being applied to fractions.  She states that her class is very much involved with sports and she feels that adding a context to fractions will help them understand greater.  She sets up her fractions with a basketball context and relates the shots made over the total shots taken.  For example, if a player made 7 shots out of a possible 10 in one game then they would have a field goal percentage of 7/10 or 70%.  She then moves into the process of solving proportions and cross multiplications all while still using the same context.  I believe that this is a really creative idea to use as students can use real NBA or WNBA player statistics while still understanding fractions clearly.  The students can then answer the question as to "why" themselves as now when watching basketball they can apply their fraction knowledge to real life NBA or WNBA statistics.