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.

Smartboard: Hit or Miss?

This week in class, we looked at the importance of technology within the mathematics classroom.  Education has greatly changed over the past 20 or so years with the use of technology.  When I was in school, the only technology we had was about 10 computers and a few televisions.    In today's integrated classroom, technology can be used everywhere from online games to iPads which enhance the learning of students and engages the class as a whole.
Although there are so many technological tools to enhance learning, we as future teachers need to make sure we are using it appropriately and effectively in the classroom.  Often times, we can overuse technology to the point where it seems repetitive and uninteresting to the students.  An example of this that we discussed in class was the idea of smartboards.  When these first came to the teaching profession, there was a large push for them as they were marketed to replace the boring chalkboards with an innovative and interactive learning tool.  Basically, smartboards are essentially a touchscreen whiteboard computer that students and teachers can write on, move objects around and point and click with the touch of their fingers.  Smarttech.com states that it is a complete "learning system that combines award-winning technologies to produce an interactive whiteboard".  Seems pretty innovative as clearly you could integrate this in a lot of lessons but we have to remember that these smartboards are not cheap!  In fact, according to smarttech.com, these whiteboards range from $3000-$4000 dollars.  Many classes both in highschool and elementary school do have at least one smartboard in the room which means a good portion of the school's money has already been spent on these boards.  This would be fine if the smartboards received great praise for being an extremely innovative tool however many educators feel that it just really isn't worth the investment.  In fact, Lisa Nielsen a teacher in the United States suggests that smartboards don't change the model that's broken they just make that model way more expensive (http://theinnovativeeducator.blogspot.ca/2010/05/why-smartboards-are-dumb-initiative.html).  Essentially, she feels that even with a smartboard the teacher still controls the content, stands in front of the classroom and has to manage a bunch of kids through a lesson plan they'd rather not be managed through.  She feels that a lot of teachers are using the smartboards to teach traditional lessons because they do not have the required training to fully unleash the capabilities of the smartboard.  I myself do agree with this statement as I too have witnessed a teacher using the smartboard as a basic whiteboard.  While I was doing an observation placement on the first day, the teacher was telling me all about the new smartboards in the classroom.  When I went to observe his lessons all he did was put a powerpoint on the smartboard, stand at the front and talk for 40 minutes and barley use any of the vast capabilities of the smartboard.  Judging from the reactions of the students around me I could tell that the lesson got very dull quick even though he used the smartboard.
Overall, I feel that smartboards can be an extremely useful tool because it has so many capabilities.  But, for the price I feel that every teacher should be trained on how to harness these capabilities and create a creative and fun learning environment for the students.  I have posted a link below which shows some cool things you can do on a smartboard.  Let me know what you think, are they innovative ideas or not so innovative?

https://www.youtube.com/watch?v=T0IGg0UaN3Y

Engaging Diagnostic Assessments

This week in class, we learned about the term diagnostic assessment.  Essentially, this can be defined as an assessment that provides educators with information about each student’s prior knowledge before beginning instruction (smarttutor.com).  Usually, the best time to implement this type of assessment is at the beginning of a lesson or unit.  The assessment essentially tells the teacher where the students are at in terms of their prior knowledge and any potential gaps or barriers that might be in the way.  Often times especially in math, having the foundations are essential to understanding greater mathematical concepts as they build on each other.  For example, to understand collecting “Like Terms” in algebra you must know the basics of BEDMAS as it is important for achieving the correct answer.

                There could be several reasons why a teacher wants to conduct a diagnostic assessment.  First, often when students come back from the summer break into a new grade, it is important to find out where every student is at in terms of their mathematical level.  This will help the lesson planning process be a lot smoother if you can gauge where your students are at.  Usually, at the beginning of the school year a teacher will conduct a diagnostic assessment over the first few days of class.  Using this information, they can review or change certain topics in their lessons to provide a better learning experience in the future.

                There are several different types of diagnostic assessments that can be used in the classroom.  Often times, there is a worksheet or quiz covering last year's material that is handed out which acts a refresher for the students  Although this is a good idea, there are much more interactive diagnostic assessments to get students excited for the upcoming math year.  One idea that was discussed was the idea of using an online application called Kahoot.  Essentially, this application works by the teacher inputting several questions of their choosing and four multiple choice answers.  When they create a game, students are able to connect to it by using their phones, tablets, laptops etc. and instantly join and play.  They are awarded points based on whether their question is right and how fast they answer.  It is an extremely interactive game and would be a great diagnostic tool as the teacher can review the results after the session.  I have provided a quick link to video on how the game Kahoot works. Check it out ! I highly recommend it
                In conclusion, diagnostic assessments is a great tool to get an understanding of students previous knowledge.  This becomes very important at the beginning of new units or even the first day of class as it allows the teacher to gauge where their class is at.  Using interactive diagnostic assessments such as kahoot keeps the classroom fun and engaging for the students as well as provides the teacher with appropriate feedback.

https://www.youtube.com/watch?v=pFFv6_6was4

Differentiating Learning (Instruction)

Differentiating Learning is an essential idea that must be implemented into lesson and unit plans in order to create an effect learning space.  Differentiating Learning or (DI) can be defined as “Differentiation means tailoring instruction to meet individual needs. Whether teachers differentiate content, process, products, or the learning environment, the use of ongoing assessment and flexible grouping makes this a successful approach to instruction”( http://www.readingrockets.org/article/what-differentiated-instruction).   Basically DI is important as every student learns at their own pace, style etc. and by tailoring lessons we can reach every student’s individual needs.

In class we observed three different scenarios that related to differentiating instruction.  We then had to analyse the scenario and identify areas of formative assessment and differentiated instruction.  Each scenario was unique but they all based around one teacher and a math class.  Some of the ways they were able to differentiate the lesson was having students use electronic resources such as an online game called “CLIP”.  Here, each student can use this online tool to meet their specific needs.  For example, a student who is strong and confident in math is able to complete the lessons at perhaps a faster pace or at a slightly higher difficulty.  Moreover, a student who perhaps came from a different educational background can understand their lesson at their own pace that feels comfortable to them.  This eliminates confusion and poor student engagement in the classroom as students individual needs will be met which is not always what happens in a traditional education lesson. 

            In observing the picture above, it can be seen that there are many different types of students in a classroom and obviously not all of them learn in the same manner.  This is why differentiated instruction is important to attempt to meet individual needs.  For example, a student who perhaps just immigrated to Canada may not have English as their first language.  This means that in a math class, not only do they have to learn the math material as everybody else they also have to try and understand it with their second language.  This can become extremely hard for the student as they would have to learn the language and material at the same time .  Moreover, this may lead to them falling behind and feeling overwhelmed in the classroom.  I know there are math lessons and activities online that you are able to change the default language which could make learning math easier to the student in the example.  I think this would be a great idea for students that may have different language abilities. I often look at myself and think how hard it would be for me to learn math in another language as it can be a difficult subject at times.  Differentiated learning is critical to the lesson planning process as it could help students with different needs, backgrounds, language abilities, cultures etc. learn the subject in their own style and pace.