This past week in EMTH, we talked a lot about lesson plans. From what consists of lesson plan to looking at the curriculum of different math classes that are taught in high school. This got me thinking of how I will use what I learn in this class in the classroom when I teach.
The first thing I have learned is that it is a good idea to give a variety of questions for assignments. For instance, it is a good to start with questions to make sure they understand the concept. Then, there should be problems that I talked about in a previous reflection, which were questions that make math problematic for students. These questions can help students put everything they know to use and make them have to really think about how to arrive to an answer. These questions can be solved by using different strategies by students which may not be used if the questions were easier. Overall, I think it is important to challenge students even if it sometimes frustrates them because at the end of the day it will help them learn.
One thing that I need to remember is that everyone is different and therefore will have different thinking processes. This will apply to the classroom because students will sometimes have different methods of solving problems. It may be different then what I will teach but still be right and as long as they explain how they got their answer I think it is good to have different methods. In the textbook teaching Mathematics through Problem Solving Grades 6-12 written by Harold Schoen and Randall Charles it states: “ […] the best way to gain deeper understanding of a subject is to search for better methods to solve problems.” I think that is a good way to look at this because with all the problems that we have done in this class I remember them because we looked at a lot of different methods to solve the problems. I found that this was also helpful in different high school and university math classes because when we looked at different ways to come to the solutions to problems it helped me understand them better.
Those are just two examples of what I have taken from my class so far. I have learnt a lot from this class that I will use in the classroom from lesson planning to teaching. The examples I have talked about in this reflection are some things that I think are very important which is why I wanted to talk about them.
The following reflection will focus on allowing mathematics to be problematic for students. This was a section in Chapter one of teaching Mathematics through Problem Solving Grades 6-12 written by Harold Schoen and Randall Charles. This reflection will be about the reading and how this relates to my own experiences with teachers choosing questions with different degrees of difficulty.
I think some people may struggle with the idea of making math problematic for students. Whether it is parents, teachers or students who do not think math problems should be problematic. But the textbook states the following: “Allowing mathematics to be problematic for students means posing problems that are just within students’ reach, allowing them to struggle to find solutions and then examining the methods they have used.” I think some people may associate problematic with impossible but that’s not the case. Students should be given difficult math problems that are not impossible but at the same time makes them apply everything they know and makes them have to think a lot about the problem before arriving with the answer.
There is a fine line between questions being too easy and too difficult. I have come to the conclusion from different teachers I have had for math growing up and to this day. For instance, I have had questions on math assignments that were a lot more complicated then they needed to be. It went beyond making math problematic, it was unnecessarily difficult to the point where nobody in the class could get the question. On the other hand, there have been different math questions that were too simple, it did not take any effort to come up with the correct answer and did not challenge me in anyway. I have had questions that are a happy medium though. Questions that make me a little frustrated and take a lot of time but I can come up with the right answer after applying my previous knowledge and using different strategies to come up with the right answer.
I think it is a good idea for teachers to challenge their students by giving math that is problematic to help their students understand math. I find myself that a lot of my knowledge and understanding of math has come from math problems that are problematic. In conclusion, I think that although it could take some getting used to but allowing mathematics to be problematic could greatly benefit students.
In this reflection I will be taking a look at the assigned reading for this week and give my thoughts on a particular section of the reading based on my feelings and personal experiences. This reading is from the textbook teaching Mathematics through Problem Solving Grades 6-12 written by Harold Schoen and Randall Charles.
In the first chapter, it gives two reasons why understanding math is important. The first one is: “[…] understanding a topic ensures that everything one knows about the topic will be useful. One will remember things when one needs them and will use them flexibly to handle new situations.” I agree completely with that statement. For instance, different concepts that are learnt in math one year can come up again many years later. For instance, in grade four I learnt how to do long division. We did enough examples that I understood the concept and even though I could use a calculator for long division for most grades after that, I still knew how to do it. Then, in my first year of university long division came up again but instead of just numbers, there were polynomials. So, I remembered how to do long division but I had to put my prior knowledge to use in a new situation. This example demonstrates what the first reason of why understanding math is important. That is just one example though, the book gives a different one but there are many more than just those.
The second reason is the following: “[…] to set understanding as a central goal.” I agree with the second reason because I understand how it feels to actually understand something compared to just memorizing a concept or barely getting by on a concept. When I am learning a new difficult math topic and having troubles with some of the questions from the new topic, it is frustrating. But then, when I work through the questions and finally come to understand the questions and topic, it is very satisfying. I have been in situations in Chemistry or Physics though when, in some cases, I have just memorized what to do without really understanding the concept. It is nowhere near as satisfying then understanding a concept. In general, it feels a lot better when I can talk confidently and fully understand different concepts than just memorize what to do and keep being frustrated. Therefore I agree why this is the second reason they give in why understanding math is important.
This was part of the reading really stood out to me because I could recall on my own previous experiences and make connections to what the main points they were trying to get across.
This week, in a second year course I’m taking called EMTH, we covered a couple word problems in class. There was an example called the Paper Strip that we did. It is from the textbook Thinking Mathematically written by J. Mason, L. Burton, and K. Stacey. The question is the following:
“Imagine a long thin strip of paper stretched out in front of you, left to right. Imagine taking the ends in your hands and placing the right hand on top of the left. Now press the strip flap so that it is folded in half and has a crease. Repeat the whole operation on the new strip two more times. How many creases are there? How many creases will there be if the operation is repeated ten times in total?”
We were then asked to do the problem. To complete the problem I decided to take a piece of loose leaf and test the question out using the paper because I find that using paper as a visual helps me. I found out the following pattern from using the piece of loose leaf:
# Of Folds: 1 2 3 4 5
# Of Creases: 1 2 4 8 16
The pattern is that the number of creases is multiplied by two with every fold. So, to solve the question I added 1+2+4+8+…+256+523=1023. So, if the operation was repeated ten times, there would be a total of 1023 creases.
I then started to remember that I did this sort of problem in high school. In Pre-Calculus 20 I remember doing a unit on geometric and arithmetic sequences and series. I then looked at the Saskatchewan Curriculum online and looked at the outcome and indicators that are involved with the unit to see where this problem would fit under. It would be under the Outcome for P20.10 which is: Demonstrate understanding of arithmetic and geometric (finite and infinite) sequences and series. This problem is an example of a geometric series and so I would use this problem in the classroom as a problem for my students before introducing geometric series to them. I would have my students do the problem to see how my students would get their answer and after discussing how some students got their solutions I would introduce geometric series. Then after teaching them that there is a formula for these types of problems I would have them do a similar problem and apply the formula.
I really enjoyed doing this problem in class this week because I was able to connect this problem to things I learned in high school and then was able to figure out how I would use this problem in a future lesson plan.
Learning from Our Students is an article written by Nel Noddings from ProQuest Education Journals from the summer of 2004. This was a required reading for my ECS 300 class this week. The following post will be my response to the article, specifically including quotes that I can relate to from personal experiences, my teaching philosophy and how I will use the ideas in this article in the classroom.
In the article Noddings states, “the true pedagogical task is to illustrate the full potential of the subjects that students are forced to study.” I think this relates a lot to what my education philosophy is. Not all students are going to enjoy the different subjects they are taking and I think as a math teacher it is part of my job to have students see why I love math so much. If students see the passion I have for the subject it is possible they can see the full potential that math has. Even though students may dislike math and but have to take the class, I can do my best to demonstrate all the wonderful thing math has to offer.
In the closing paragraph of this article Noddings says “Additionally, through listening to students, we can learn what motivates them, and we can pique their interests by sharing our own intellectual interests.” I would like to bring this concept to the classroom. I think it is a good idea to listen to students because I agree that teachers can use a lot from students. When teachers know different learning styles and different things about students we can help them learn. Also, when we share our own interests with students we may have similar interests or spark interests students did not know they had.
Overall this was an interesting article that brings up different points that I may have not thought of. I would recommend reading this article because it is an interesting article that gives a lot to think about.