Equivilant Fractions

Idea: Different look, same value

Reflection 1: I understand the idea about equivalent fractions well mainly because I understand how things are related and match up. What this means is, I can see how there parts of a whole and if you have the correct number of these parts to make a whole, than you have made a whole.

Reflection 2: Many children may not feel the same way as I do when first learning about equivalent fractions. For a child with a fairly concrete mind, once they are told that a whole is a whole number (3, 4, 5, 6 etc), than it is challenging for them to understand that there are many pieces to the while that they do not see (halfs, quarters, thirds etc).

Reflection 3: I'm glad to see that I am thinking about student's developmental stages as a student teacher. I am a psychology lover and have a large background in psychology (and eventually would like to continue in this field), therefore it is important to me to really see the cognitive, behavioral and physical stages these students are at.

There's More to Math

Idea: We do things, without knowing we do it.

Reflection 1: This comment from math class a couple of weeks ago was important to me because it not only applies to just math, but to many things we do every day. These are the things that are so second nature to us. I think that it is important for math to become something second nature because math is all around us, whether we notice it or not. We do small calculations, estimations, find patterns, and process a lot of information at once in our heads as well as multi-task in our environment. There are a lot of variables around us, as well as statistics and spaces. Learning new math concepts works the brain. It takes a while for these things to become second nature. I think that is why learning time is so precious for these understandings to ground themselves. For students to be successful they need to practice and rehearse these new things. For example, I still need to use my fingers to calculate 6x8. Many people can come up with the answer automatically. It takes practice and exposure to something many times until it truly sticks. It's like learning another language. When you are not immersed within the language it makes it a whole lot harder to become fluent.

Reflection 2: I definitely want a fluency for my students, and I want a fluency or myself. After my first practicum, I felt that I would never again have fluency, a sense of confidence in my practice. I think you can always start over when you learn. I know that we are life long learners. Once we stop learning, or chose to stop learning, I don't think it is impossible to begin a learning journey again. You may be bruised a little, but it's not impossible. For one student in my last practicum, I think the above idea didn't really apply to her new adjustment to the classroom. I think perhaps she didn't do things, without knowing that she could. She needed stimulus, someone to lead her. I don't think I did enough to help her, but I recognize now that there are many sides to a learner and encouraging them to enrich themselves in the lesson can't hurt.

Kicking the Habit - Textbook Trouble

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Idea: Textbooks are trouble

Reflection 1: Like many teacher resources, textbooks guide students through their learning of concepts but for some reason textbooks are missing the boat on the way students learn and providing a way for a teacher to use the resource effectively and efficiently. I say this because many unnecessary concepts are introduced at inappropriate age levels as well as man examples are insufficient. I don't necessarily think that textbooks are complete trouble, more so they are an option. One that can be a crutch to many teachers who rely on it because they are lacking skills in the area of math. During my practicum I would say that I relied on the textbook because purely a lack of expertise. The textbook was available for me to use - and of course I took advantage of the information inside.



Reflection 2: I seem to be very back and forth about the use and perhaps abuse of textbooks for teaching. When I used the textbook I would get confused while was teaching. I tried too hard to have the students follow what was in the book . I'm not really a learner who learns incredibly well from reading (I have now learned) I learn better when I see it happening, it demonstrated. I think this is just what I'll need to do with my students.The students need to do more inquiry instead of simply believing what the textbook says to be true. I think it's more important for students to find that out themselves. S0 ... I guess no textbook.

The Secret Key

Idea: The secret key to mathematics is pattern.

Reflection 1: I found this line on a math website a while ago while I was trying to look for math lessons and it seemed to have stuck in my head not because I love it, find it inspirational or truly understand it. I think it stuck with me mostly because it is so definite, clear and confident. Now that I think about it, to better understand what it mean, I first need to look it up. What is pattern?

There were 5 definitions on www.dictionary.com. I've listed the 3rd, 4th and 5th because I feel they are most appropriate for math.

3. a natural or chance marking, configuration, or design: patterns of frost on the window.

4.a distinctive style, model, or form: a new pattern of army helmet.

5.a combination of qualities, acts, tendencies, etc., forming a consistent or characteristic arrangement: the behavior patterns of teenagers.

I like these definitions. Math is a design. It's a group of 'things' together in some kind of arrangement but in a distinctive, specific style. Math has a meaning. Numbers are woven through, stitched, and placed next to each other to create something new. It's consistent and reliable. Sure sounds a little romantic doesn't it? A little artsy fartsy? I thought so too. But, I like thinking about math this way. It's approachable for me this way. I'm a drama student and spend most of my first year at university making jokes at the Faculty of Engineering's expense.

I am feeling closer to this "secret key" notion. In math there is a series. We add, then add, then add again to multiply. We take away, take away and take away over and over again, to divide.

Reflection 2: I have a memory for moments when I have been moved towards an idea, thought or word of wisdom. Mathematics is not my strength, I've noted this multiple times already in other posts and having this connection to a pattern and a variety of accessible definitions, helps me to understand mathematics as an art. It's some kind of production you could say.

Think About Why!

Idea: Real mathematicians ask why

Reflection 1: It's important to me and as I facilitate student learning to create an environment for students to question the concepts I am presenting as well as question the way that they think about this new information. When it comes to math, however, I don't think that students are really given the proper opportunity or the permission to ask higher level thinking while learning math. I found in my last placement that the questions students were asking were procedural, to repeat instructions or just for reassurance that what they were doing was correct. There is nothing really wrong with those questions, but perhaps it is the territory that comes along with "math". Unlike in the literacy block, teachers do not ask many "why is that?" questions to students to explore, or dig a little deeper as to why math things are the way they are. For example to help students predict, invent or problem solve, teacher may ask, "what would happen if...?" or give the students a chance to "convince me...!". Rarely do we also allow students to express true feelings, attitudes and beliefs about mathematics. In literacy we do. In science we predict, why not in math? Why not express our attitudes towards math so that we are more comfortable and confident with the subject?



Reflection 2: I want to be this kind of teacher, able to provide students with prompting questions about math and open them up to deeper thinking rather than doing simple work sheets. I made this mistake in y first placement. I think I know better now.

I Met Math On eHarmony. It Didn't Work Out

Idea: If you are going to teach math, you have to enjoy it!

Reflection 1: If you are being forced to teach math, you better pretend to enjoy it!



Reflection 2: It's difficult as a teacher candidate to teach a subject that you yourself are not completely comfortable with. This comment if fair too commen - yes I understand that I'm preaching to the choir - but for someone who knows what to do, but just isn't sure HOW to so it, it can really make an experience difficult. I will be the first one to admit that my abilities to teach math are not acceptable - but I'm trying. How am I trying? I'm pretending. During high school and some parts of my undergraduate degree, I learned to "fake it till I make it". Sometimes if you fake it enough, you actually start to believe it. But, how do you know when enough is enough and I should "stop digging myself a grave" or 'quit while I'm ahead". I just don't want to "go down in flames" again...

"Sorry Miss. I Can't Do It. I'm Waiting for a Calculator"

Idea: Not only slowing down the Math Stations, some students in my last practicum (Grade 7) depended highly on their calculators and it was surprising by how much they saw the legitimacy in this even if the problem was age appropriate and fairly straight forward (adding and subtracting three digit numbers)

Reflection 1: From what I remember, when I was in school we used calculators. We definitely used them in Grade 8 when finding the area of a circle, and I wouldn't doubt that while the calculator was on our desks we would type in simple problems that we could have done in our heads. It was convenient. But, it's also a tool that was invented and allows us to do these "simple and silly" problems. You can't really blame the technology.

Reflection 2: But, my kids were WAITING to use a calculator - for their friend to finish and pass it over. While waiting, students would get social and not even try to attempt the problem without my prompting. Is this a norm the education system has created? Like putting up your hand, or lining up at the door before you enter the room? Unless your teacher says not to, use a calculator! It's easier.

Miss, You're Going Too Fast!

Idea: No matter how long it takes, if they haven't mastered the skill you just can't move on.

Reflection 1: I agree with this whole heartily, but I have this feeling that it would be hard to actually practice this pedagogy in my own classroom. I for sure had this problem in my first placement. My associate kept telling me, "you're going to slow", "you need to hustle" and even went so far to tell me to teach three lessons in one 50 minute period. There HAD to be a student that was behind after that class. And I think I knew who it was, and of course I don't think I did enough to help her out. The amount of have, I didn't have enough of it

Reflection 2: I think as far as my teaching practice and presence in the classroom goes, I need a bit more time myself to fully grasp the important things that are going on. Math as a whole I'm not comfortable to teach to start, so not only will students need time to grasp concepts I need some time too. I think that's where I had the most problems in my first placement. I needed time too. I needed some sense of mastery too.

The Attitude of Measurement and Muffins

Idea: There is an attitude to measurement. "I'll be back in a jiffy" doesn't ring as tightly as "see you in an minute and a half". There isn't really an obligation with the words "jiffy" "pinch of salt" or "just a second", but when we measure and things we are measuring are required to be in a certain amount, we make ourselves at risk of failure. I would get into trouble if someone expected me a minute and a half ago

Reflection 1: When it comes to baking, I'm not that much of a stickler on the measurement of ingredients for desserts that I know are going to turn out alright anyways. For example, this Cinnamon-Sugar Pull-Apart bread came out wonderfully delicious and all ingredients along the way were not as precise as they really needed to be. For example the online recipe called for 2 3/4 cups plus 2 tablespoons of all purpose flour. I don't know about you, but that is an odd measurement. Leaves a lot of room for error. Again, the tasty treat still came out of the oven smelling just as I thought it would. Muffins, however are a different story. Over the past month I've baked 2 different kinds of muffins. Carrot and blueberry, both calling for specific amounts of ingredients including baking soda and/ or baking powder. I've always thought of these ingredients as high maintenance, temper tantrum possibility ingredients, therefore I try my best to not mess up the steps in the making of muffins. It's odd (1) that I feel intimidated by something purchased at the Bulk Barn, (2) the more I try to get it right, the more dirty dishes I have, and (3) that when I feel successful, the muffins prove me wrong and come out rock hard and inedible. Trust me. Inedible.

Reflection 2:
I'm not making anymore muffins. They intimidate me too much with their fancy specific measurements. I think the attitude comes from our perception of the thing you are baking. Today for example, the Cinnamon-Sugar Pull-Apart Bread looked amazing before it went into the oven. I could have eaten it right then in there - all I had to do was simply "heat it up first". Muffins are not like this at all. There beginning state is inedible, liquidy, eggy, and (not the good kind) gooey. It's the oven that transforms it.

Reflection 3: You can't trust an oven.

Long Division Smells

Idea: Why teach long division if no one uses it in real life?

Let's Look it Up!: From R. James Milgram of California State University Northridge at a 1999 Conference on Standards - Based K-12 Education (http://www.shearonforschools.com/why_long_division.htm)

Skills Directly Associated with Long Division:

• Students cannot understand why rational numbers are either terminating or (ultimately) repeating decimals without understanding long division
• Long division is essential in learning to manipulate and factor polynomials
• Polynomial manipulation and factoring are skills critical in calculus and linear algebra: partial functions and cannonical forms

Long division is the only process in the K - 12 mathematics curriculum in which approximation is really essential. The process of long division is a process of repeatedly approximating and improving your estimates by an order of magnitude at each step. There is no other point in K - 12 mathematics where estimation comes in as clearly and precisely as this. But notice that long division is also a continuous process of approximation, the answer keeps getting more and more accurate and when the students learn how to do long division with decimals they learn to carry the process to many decimal places. This leads naturally -- in a well conceived curriculum -- to students understanding continuous processes, and ultimately even continuous functions and power series. The development of these skills are all contingent on a reasonable development of long division. I don't know of any other or any better preparation for them. (Milgram, R James)

Reflection 1: Long division seems to be a foundation for a lot of math that students will be learning in high school and university level classes. I really shouldn't be the one to neglect them of this material and this knowledge. Although I am not a large fan of the process that long division brings, it will help students to understand more that is to come. Like anything worth doing, it will take time.

Reflection 2: I am willing to give long division a chance from reading the transcript of Mr. Milgram's speech. He seems to be passionate about the subject, so I feel I should to.

Reflection 3: When I don't know the answer or am unable to sound intelligent on a subject, I'll Google it.

Reflection 4:Of course I knew all along the importance of long division!! It still smells, but I'll do it.

Milgram, R. James. "Why Long Division." College of Science and Mathematics. Northridge, California, 1999

Makes Me Think of NOLA Again

Idea: O is a number, Nothing is a quantity

Reflection 1: The word 'nothing' has the ability to punch me in the stomach for a couple of reasons. The main one I believe is that I have seen what 'Nothing' looks like. I have heard what 'Nothing' sounds like and I have felt the atmosphere of 'Nothing' when I visited New Orleans's Lower 9th Ward in February, 2009. It's incredibly heart breaking visiting a city that has so much excitement, musicality and familiarity but once you remove the band aid for a short time, you see its truth and the truth of a country with whom turned its back on such a brilliant place. This place is still hurting, at some points on my visit were numbing and I felt waves of shame. The 'Nothing' I speak of is abandoned homes that once held a family. Their Christmas dinners, graduations, and birthdays are without a place to celebrate. Empty schools, restaurants and streets litter the city. Nothing is an amount, which can be mistaken as a number, but O is a number, therefore it cannot be an amount. Is this true? I know that O is represented on a ruler, therefore it exists, I also can get a ticket for not having my seat belt on when my car is idling. O is a speed. It's a number. 'Nothing' is deeper. 'Nothing' effects people, changes people, moves people to find more than 'Nothing'. One thing I must say is that New Orleans is being rebuilt, and the people who had experienced Hurricane Katrina and decided to have faith in their home have celebrated its roots and strength through adversity. For people who once had nothing, they sure have created much more from their situation, and perhaps as I think about it now, were never without. Their memories were always intact. They still had music in their hearts. Just because you see 'Nothing' doesn't mean 'Something' isn't there. 'Something" is a quantity too isn't it?

Reflection 2: Math helps us understand our World. The World helps me understand math.

Disassociate From What We Know

Idea: In order to teach well, we need to disassociate ourselves from what we already know and explore something new

Reflection 1:Many of us were taught adding, subtracting, multiplying and dividing in a certain way in school. Now as future teachers we need to start again from scratch and understand that there are many ways to solve math problems, and that as long as you get to the answer, your way is correct. Because our students will be learning at different rates and in need of a variety of ways in which material needs to be presented, we as teachers need to be flexible to this so that our student's are the most successful. Although I was met with some hesitation in learning these new ways, I knew that I needed to think a new way in order to help my students.

Reflection 2: I will eventually need to change some of my student's minds as well and help them explore a new way of learning math from what they experienced in the previous year. I know that it's easier to live within what we know even if it's wrong, so I might need to have some support when the time comes. Or just be really really persuasive.

Rearranging the Question

Idea: Think of the problem the opposite way to understand what it is asking.

Reflection 1: The way I initially looked at this problem 123 + _____ = 300 was to take 123 away from 300 in order to find the difference. This would be the blank area. Although this is a quick and simply way to find the missing value, it is not what the question asked. 123 + how many more = gets us up to 300. This would be a long and tedious method of solving this problem, we had to add more until we reached the desired number. 300. This was frustrating because the ______ served as an unknown. The value after the = should always be the unknown, therefore I rearranged the question so I could feel comfortable trying to find the end value instead of the middle.

Reflection 2: I am more comfortable in knowing where I am going instead of how I'm going to get there. Maybe I should reconsider.

Multiplying

Idea: We can use Base 10 Blocks in a grid to help us multiply numbers

Reflection 1: Blocks take the place of numbers in the grid, but we need to break down the number into par to make it easier to work with them. For example, 26 X 16. Instead of 26, separate it into 2 tens and 6 ones and instead of 16, separate into 1 ten and 6 ones.The block on the outside, frame the products we are creating inside. Eventually there will be more blocks than what we started with.

Reflection 2: When we multiply, we always get more.

Learning With Things

Idea: Today we learned that when students manipulate concrete objects such as base 10 blocks, their learning and development of meaning is linked, therefore achievement is enhanced.

Reflection 1: We know that student’s learning should be pulled from all senses in order for them to truly experience learning. This is why we recognize and take into consideration auditory, visual, and kinesthetic learners as we create our lessons. I believe that doing math should not be any different, and would hope that within my own teaching I take advantage of the pattern blocks, base 10 cubes, sticks, and other math tools in order to help my students reach curriculum expectations with ease.

Reflection 2: The math blocks and other tangible math tools are putting into our own hands what we know and what we need to see in order to truly understand. Math is about discovering, counting, measuring, evaluating “things”. I want to helps students use these higher levels of thinking and be able to concretely understand how things come together or are divided. I would feel validated as a teacher if the students would be able to teach one another concepts as they manipulate blocks, perhaps even help their parents see what they are learning.

Are We Taking Numbers For Granted?

Idea: Today we learned that we can use symbols for numbers to represent things other than numbers. Computer language (binary 011001100101010) or slang/ text message language (b4, 2day, gr8), phone number 226-555-1234, username smellycat_08, and social insurance number 345 232 786 are all examples of how we use the symbols for numbers, but are used in a way that do not require their respective values to be used. I am not the 8^th smelly cat in Ontario, nor am I 345 232 786 years old. The numbers are not place holders, nor do they represent a time or age.

Reflection 1: Are we taking the shapes and sounds of numbers for granted, therefore abusing their significance? Or are we adapting our view of what numbers represent into a technological, fast paced society?

Reflection 2: I know that language changes quickly, so why not numbers? If math is a way in which we try to understand our world, and written or oral language is a way to tell people what we know, than why can’t they work together in which to change and adapt as technology does? I remember when text messaging was first developed. I was charged for every ‘character’ (number, letter, blank space, exclamation point, period etc.) I used within one text message. Therefore, it was more economical to shorten phrases so that they were still able to communicate. For example, the times I would meet someone or how boring my biology teacher was. The students of this age, including myself innovated a new language by combining the sounds of letters and the written numbers (two, four…) to create sentences. Using this new language, I do not consider myself as illiterate, rather I am changing language and numbers to fit my life, instead of language and numbers leading mine.