21 IEP's

Before the start of my first placement, I called up my Associate Teacher to discuss the class I would be teaching in. We spoke of many things, but what stood out for me was the number 21. The number 21 was not the number of students in my class, there were going to be 31 students. 21 was the number of IEP's in my class. IEP means a student with and Individual Education Plan. I told a few of my fellow classmates that I would be teaching in a class with 21 IEP's they were all scared for me. I spoke with my mom who is a grade 4 teacher, and my step-father who is a retired principal..... they were also a little nervous for me..........
I was a little nervous myself.








I was barely able the sleep the night prior to the beginning of my first placement. I kept dreaming that I had 21 wild animals running around my classroom.


So I started my first day, sleep deprived and with a large lump in my throat.






At the end of the first day I had come to the realization that the number 21 meant absolutely nothing. If anything I should have spent more time thinking about the number 31. This is because I had 31 unique boys and girls all eager to learn and participate in their own way.  

Over the course of the 4 weeks that I was in teaching these students, I developed an deep appreciation for each and every one of them.  The IEP's were a bonus. They let me know that these students learn and act differently, than what society would determine the norm.  I however, considered there to be 31 IEP's in my classroom. Every student is an individual, and should thus be aloud and applauded for being such.

3 part math lesson

In my first practicum, I was teaching grade 7/8 Mathematics.  Let me tell you..... It was absolutely AWESOME. Children are incredible, they are sponges that just want to learn. I was fortunate enough to have an Associate teacher who demonstrated to me the wonder of a 3 part math lesson. The following is the criteria for a 3 part math lesson.

I thought teaching math was going to be a drag, I thought that I would be up lecturing for 10 to 15 minutes and then I would assign the students questions from a text book. They would have the rest of the class to complete them and if not done they would be for homework. I recall as a student not understanding a certain concept, others in my class would also be lost and the teacher was unwilling to help. I believe that the teacher had a deep disdain for mathematics and could care less if her students missed the boat. She would just move on and teach the next lesson. I thought that this was the way math was to be taught, and I was a little uneasy when the 3 part lesson was first brought to my attention.

After conducting my first 3 part lesson, I felt like Moses. The seas of understanding parted way for me.

The students were able to interact and help one another throughout the class. Reflecting, Discussing, Observing, and Investigating are just a few of the many ways in which the students interact with one another when learning. I, as the teacher was able to walk around, answering group questions after they had discussed their findings as a team. Giving little prompts when necessary.



The final part of the lesson; consolidation was far and away the most rewarding part of the lesson for the students as well as the teacher. The students were able to present and apply the knowledge that they had learned. They were all little teachers themselves, teaching the rest of the class along with me, the manner in which they came to their answers. This was amazing, because the students were able to demonstrate their findings in ways I hadn't even thought of. This benefited the entire class, for it was a way of teaching some students who had yet to get the entire grasp of the concept being taught, another way to learning it. It also re-affirmed the lesson for a number of students.


In the end the 3 part lesson, should and is the new baseline for mathematical instruction. I have become an advocate of the 3 part math lesson. I will from here on out teach it! Because, I am not the only teacher in the classroom, each and everyone of my students becomes a teacher thanks to the 3 part math lesson.

Multiplying by 11

I have always been intrigued by the number 11. There has always been some unspoken magic regarding this number. I feel as though, more than any other time, I shoot a glance at the clock and it just so happens to be 11:11. I always make a quick wish in my head when I see this time. I have been unable to explain exactly why………

In class on Tuesday, Professor Antosz began speaking of the trick to multiplying by 11.  I for one had never heard of any such trick and was immediately intrigued. The good Professor went up to the blackboard and did a simple multiplication of 14 multiplied by 11. He wrote the number 1  4 on the board. Then with arrows below each number (1) and (4) multiplied them by 1 and then wrote them slightly spaced out. 1       4. Professor Antosz then placed a small + sign between the original 1 and 4, to look like this 1+4. The Professor took the sum of these two number and placed that number (5) in the space between the 1 and 4, to look like this; 1 5 4. BOOM!!!! This is the correct answer to 11 times 14. 

I was amazed, finally the number 11 and all its magic was showing its worth. Unfortunately, being as I am, incredibly pessimistic I wasn’t convinced. I needed more proof. I needed to see this work with larger numbers!! I decided to try, 8738 multiplied by 11.

 

I began by multiplying the first and last numbers by 1 and placing them at opposite ends. Next step was to add the numbers side by side together. I had to carry the 1 a couple times…. The result was……

 96118 the correct answer for 11 multiplied by 8738. Colour me impressed!




Looking back on the the outcome, the trick to multiplying any number by 11 works perfectly well. However, it does not work well with any other number. I played around for quite sometime seeing if any other number possessed the magic that 11 does. I unfortunately came up short.




In conclusion, the trick to 11 is great, fabulous, wonderful,..etc. But we must realize it only works for things multiplied by 11. I don't know about you, but how often in real life scenarios are we multiplying by 11? Groups are usually in 5, 10, or 12.  Even 13 when you ask for a bakers dozen. I do love the scenario for when I teach my grade 4 students their 11 times table. But any other time 11 is still 11...... That does not mean that I won't still be making little wishes when I catch 11:11 on the clock.

Is the process of subsidizing the same as counting?

When speaking to my fiancé about teaching math in her Kindergarten classroom, she introduced me to the term ‘subsidizing’ which is the brain’s natural ability to instantly recognize quantities.  In her classroom, she uses dot plates, dice and dominoes to teach this concept of subsidizing. She flashes the card and the students automatically know what number they are seeing without having to count the dots.  I really never considered this as a natural built in skill, but rather something we learn as we get older and have had experience with counting quantities or using dice.

It is very interesting to think about some young students who might not have mastered counting in correct order, but are still able to subsidize and recognize a quantity instantly. This really says a lot about our brain’s innate abilities related to mathematics. Our brains learning to subsidize naturally is a skill that can be quite useful in everyday activities. If an individual is able to easily subsidize quantities, then counting large amounts of items will be made easier by viewing items in small groups such as 5’s rather than 1’s.

I relayed this to math class with Professor Antozs; when using the 1000 units of block ten blocks though the young students are unable to count to 1000, they are able to express the number. Even everyday activities such as looking for a table at a restaurant that will hold all the members of your family can be made easier by subsidizing. Rather than having to count each chair at the tables, your brain helps you by instantly recognizing how many chairs are at each table. I would have to say that subsidizing is a phenomenal asset that we possess, it is inherently math based, which in my opinion is extremely cool! When I have students in math who say they hate math, or can't do it. I will give them an example of their inherent subsidizing skill. Then tell them that they are built in Mathematicians, they can do math!!

Division

I learned division a long long long time ago. So long ago, that I do not recall the exact manner or manners in which I was taught to divide. Whenever I need to divide something now, and do not have the use of a calculator I divide the same way as the example on the right. Seeing as I will be in a classroom in the near future. I will be the one teaching students to divide. In class, Professor Antozs demonstrated a different approach to long division, one that I was unfamiliar with. Instead of doing it the way we see here on the right, it involved taking out large groups of 1000's, 100's, 10's and singles. The division is an entirely new idea. I had difficulty at first rapping my head around it. I will illustrate an example.



As we can see in the example on the left, we are taking out large sums of numbers. Once we have taken out all of the numbers, we add them up to give us our final answer. Observing this at first, I was confused due to the increased number of steps, as well as the fact that there are now two columns. Once I looked at this process a little longer, and tried a few more examples, it became clear that this process works. It works very well!

Looking back, example number 2 does look more complex. However, when looking at step by step, it tends to be easier to follow. What I mean by this is you can see the steps and where the numbers are all coming from. In example 1, the numbers get pushed down without any real reasoning behind why?  Though we arrive at the proper answer, visually the second example leads us in each sequence without any crazy `just because` steps.

In conclusion, both systems work. They both serve well in coming up with the correct answers. The second example tends to be easier to understand. Yet if the children in your classroom can grasp example 1, there is no harm in learning that way too! As a result, I believe that opening up different learning avenues with regards to division is essential in the development of learning. I learned a new technique today. Had I been shown this process as a child there is a possibility that I may have done better in mathematics. All children differ in the way they learn. The 2nd example gives us as teachers some variety when trying to educate our students.

Sports - Statistics

I am an avid sports fan! I play sports nearly as much as I study sports! I am in multiple fantasy pools, all statistics oriented. The more and more in depth I get into these fantasy leagues, the more incentive I have to win these leagues. Not only for the enjoyment of the sport but also for the money $$$$. For example, when I am watching the Detroit Tigers play baseball I am not only routing for my favourite team to win, I am routing for the players on my fantasy team to put great statistics, while routing against certain players to put up poor statistics. When I see Victor Martinez hit a three run homerun, I am not only seeing the Tigers take a 3-0 lead. I am seeing a bunch on statistical numbers. For the week I am plus 1 in homeruns, plus 1 in runs scored, plus 3 in runs batted in, and plus a certain percentage % in both on base percentage and batting average.

The exact same scenario goes for my pitchers. When Justin Verlander tosses a complete game victory, I see more than just a 1 in the win column for the Tigers. 9 innings pitched, 1 win, 1 complete game, 15 strikeouts, 1.50 earned run average, and a 0.75 walk/hits per inning pitched.

Though the game is strictly counted in the win and loss column, there are many more numbers at play. I began to consider the money the individual players make, then the money each team collectively pays their players. I was able to find a graph that compares the average salary of the American League teams and National league teams to the New York Yankees. Over the course of the past few years the difference is astounding. The Yankees pay their players considerably more. So do these numbers pay off?



Upon further review, the salary of the New York Yankees does tend to pay off. Since the year 2000, the Yankees have won 2 World Series titles, 4 American League Pennants, 9 East Division titles and earned 2 wild card berths into the post season. The Statistics tend to point in the direction, that the more money, $$$ or numbers that you are willing to spend. The more playoff game and championships you are likely to win. Therefore, to me it seems as though mathematical numbers are equal to wins, losses, championships, dollar signs, good players, bad players, exceeding and impeding expectations.

Calories?

As I went for a run today, I was thinking about the calories that I was burning. The screen on the treadmill listed that I was burning 1100 calories per hour.  I began contemplating what a calorie was? And if I am burning these calories, it would lead me to believe that I am a composition of thousands and thousands of calories. Therefore, in the end I am a huge number.





Reflecting back, I realise that the number of calories that I am burning reflect the energy I am expending.  The calories may mean fat, weight or even sweat.  But then again calories are found in all the food we eat. For example a piece of cheesecake may contain 500 calories, so the more cheesecake we eat would mean the more calories or numbers we put in our bodies.

 

It seems to me that though the human body is a composition of organs, skin, hair, water, bones etc. We may also look at it as a number.  This coincides with food. That cheesecake may be made up of flower, cheese, salt, sugar, cream and so on. However, another person may look at the cheesecake simply as a number of calories. This leads me to believe that human beings as well as food can be viewed solely as a number.

Dividing by ZERO

Last week in class Professor Antosz wanted us to come up with a way to teach students how to divide a number by zero.  My first instinct was to search the web for the solution.  I found this video on youtube.
http://www.youtube.com/watch?v=8fBZK106C3Q
Seeing as that was no help, I turned to Google. The result was the following image.

I know that dividing by zero is undefined..... but why? When zero is divided by 3, it is equal to zero but not the other way around.

Upon further thought, the reason dividing by zero does not equate relates to the following examples.   I have 3 empty jars. These jars are empty thus they represent 0. Now I am going to divide them into 3 groups.  Therefore taking 0 and dividing it into 3 makes sense. It equals 0. I have 0 things in 3 different groups!

On the other hand what if I now took those three jars and filled them with jam.  I want to take these 3 jars of jam and split them into zero groups.  Unfortunately, I am unable to take these 3 jars and divide them into 0 groups. It is an impossibility to take a thing of substance and put it in no place.  Resulting in  3 / 0 = undefined.

Zero is a number that can be used in a multitude of ways.  It can be added and subtracted, it can be a symbol or a place, and it has the ability to create shapes and images. However, when taking a anything and dividing it by zero…. It can only be undefined!